Security scan is checking this page.
A short summary of each chapter, then the key terms that chapter names. Each chapter link opens that chapter in the OpenStax book.
Open the bookChapter 1
We're genuinely excited for you to join our community and embark on a transformative learning journey! This user-friendly guide is your roadmap to navigating the features and resources of our unique Algebra 1 content. Let's raise the bar on your algebra skills—get started with confidence today! Algebra 1 has nine units, each of which begins with a special lesson called Overview and Readiness.
Chapter 2
Use the information in this section to support and elevate your academic journey. Get tips for studying, time management, and working successfully on group projects. Explore this content and more in the OpenStax textbook Preparing for College Success , always available free online. Amplify your academic achievement and prepare for the successful future you deserve!
Chapter 3
A solution of an equation is a value of a variable that makes a true statement when substituted into the equation. To find the solution to an equation in one variable, the goal is to isolate the variable on one side of the equation. Solve Linear Equations: Mini-Lesson Review For example, in x + 2 = 3 , the value of the variable x is 1
Chapter 4
The big thing you learned is that an equation is a statement that an expression has the same value as another expression. Tell which quantities in a situation can vary and which ones cannot An equation can have letters, numbers, or a mix of letters and numbers You also saw how expressions could change if quantities changed
Chapter 5
You will be asked to evaluate the percentage of a number mentally. Sometimes, a visual is helpful to do this. Then, look for a pattern that can be used to describe how to find the percentage of any number Change the percentage to a decimal and multiply it by 200
Chapter 6
The two quantities, x and y , are related See if there is an operation that could be done to x that would produce y
Chapter 7
For a value to be a solution to an equation, it must make the equation true Most of the calories come from c grams of carbohydrates.
Chapter 8
A is the only graph that intersects the vertical axis at 0 (or that starts at the origin) C is the only graph that is discrete, showing plotted points (that lie along a line) instead of a continuous line In graphs, slopes, intercepts, axis labels, and where points are located can all create different meanings With these intercepts, you calculated the slope of the line
Chapter 9
Tell whether two expressions are equivalent and explain why or why not N 2 - 9 2 ( 4 - 3 ) or ( n + 3 ) · n - 3 8 - 3 · 2 n 2 - 9 2 ( 4 - 3 ) or ( n + 3 ) · n - 3 8 - 3 · 2
Chapter 10
Illustrate that dividing by a variable is not used in solving equations because it can lead to equations that have fewer solutions than the original equation Demonstrate that equations that are not true for any value of the variable(s) do not have solutions Through these discussions you had the opportunity to construct logical arguments
Chapter 11
You also wrote an equation in the most useful form that would be needed to find specific information In the image below, all the parallelograms have the same height. Base length is measured in inches, and area is measured in square inches
Chapter 12
You then rearranged the equation to find the number of cars and number of trucks that could be in the shipment You rearranged the equations to help solve for the number of workers that could be hired and the number of miles of roads that could be resurfaced. Solving for different variables in an equation can help you answer different questions Solving for the specific variable would help you find the number of child or adult tickets sold at the carnival
Chapter 13
You then worked on solving the equations for y and recognized where the slope and y -intercept are located in the new equations Demonstrate that rewriting the equation for a line in different forms can make it easier to find certain kinds of information about the relationship and about the graph Jada has $20 to spend on games and rides at a carnival
Chapter 14
Take an equation of the form a x + b y = c and rearrange it into the equivalent form y = m x + b You also interpreted the meaning of those numbers in the situations graphed Rewrite each quotient as a sum or a difference
Chapter 15
You also worked to find the slope and y -intercept from a graph to then write the equation of the line in slope-intercept form You then wrote the equation in slope-intercept form You then wrote the equation for a line in all three different formats when given a variety of information You then used one point and the slope to write the equation of a line in point-slope form.
Chapter 16
For questions 1 - 3, write the equation of A line with a slope of 5 and a y -intercept of 1 2 A line with a slope of 2 that goes through ( - 1 , 3 ) A line that passes through ( 2 , 4 ) and ( 1 , - 2 )
Chapter 17
Using the slope and a point, you wrote equations in point-slope form and slope-intercept form You also wrote the equation of lines parallel to given lines that went through a specific point You wrote the equations of lines that were perpendicular to a given line and went through a specific point
Chapter 18
Set up and solve direct variation problems In addition, you solved the equations for a given constraint in application problems
Chapter 19
In this culminating project, you will integrate ideas from the unit. In the first activity, you will analyze points on a graph to identify your meaning in context. In the second activity, you will identify slopes and intercepts from graphs and match equations to graphs. Finally, you will create your own scenarios to match a linear equation
Chapter 20
We connect the points with a straight line to get the graph of the equation. A graph is a visual representation of all the solutions of the equation. An ordered pair ( x , y ) is a solution of the linear equation A x + B y = C , if the equation is a true statement when the x - and y -values of the ordered pair are substituted into the equation Linear equations have infinitely many solutions.
Chapter 21
Refer to two equations as a system of equations Writing an equation is one way to check if these values meet the constraint These equations represent a system of linear equations. Then you graphed the system of equations and used the graph to complete a table.
Chapter 22
The point of intersection is labeled ( 32 , 68 ) This summer, the Farmer’s Almanac is calling for very high temperatures in central Texas. Estimate real-world solutions by graphing systems of equations You then estimated the y -intercept and slope of each equation that was graphed
Chapter 23
You determined whether a given graph could represent a specific system of equations using your knowledge of horizontal and vertical lines and the slopes and intercepts of those lines By isolating a variable in one of the equations, you found an equivalent expression that was substituted into the second equation. Using this method, the solution to one variable was found. This value was then substituted back into the first equation to find the value of the remaining unknown variable
Chapter 24
Demonstrate that adding or subtracting equations in a system creates a new equation, where one of the solutions to this equation is the solution to the system When two equations that are true are combined, the resulting equation is also true By using elimination, you created a third equation containing only one variable. This third equation can be solved to find the value of one of the variables.
Chapter 25
By determining that the equation was still true, you discovered that adding equivalent values to each side of an equation maintains its truth You then determined that adding these equations together results in a third equation with the same solution Some systems of equations are more suited to be solved by substitution and others by elimination You reinforced that to solve a system using elimination by adding, the coefficients of one of the variables must be the same value and have opposite signs.
Chapter 26
Then you created new equations by multiplying any of the original equations by the same number. After graphing these new equations, you learned that multiplying an entire equation by a number does not change its solution By solving the new system of equations, you showed that the solution to the system of equations was not changed from the original system of equations Since any one of the original equations can be multiplied by any number without changing the validity of the equation, there are many ways to cancel out a variable when combining the equations in the system
Chapter 27
When solving by graphing, the graphs of the equations are the same line. When solving by graphing, the graphs of the equations are parallel lines that never intersect. When solving by substitution or elimination, you end up with an equation, such as 4 x = 4 x , that is always true no matter what value is used This means there are no values that make both equations in the system true.
Chapter 28
You also explored the value that is at the boundary of an inequality and consider whether it is or isn't a solution to an inequality To explain what the letters in the inequalities mean in the given context, you could not simply match the numbers in the verbal descriptions and those in the inequalities. You worked to attend carefully to the symbols and any operations, and reason both quantitatively and abstractly Writing Inequalities to Represent Constraints Previously, you interpreted given inequalities and made sense of them in terms of a situation.
Chapter 29
You know that an open circle is used to represent < and > , while a closed circle represents ≤ and ≥ Adding context can help you intuit why the solutions to an inequality are often represented by a ray on the number line. The activity also prompted you to think about the solutions to an inequality in terms of a related equation The context helped reinforce why it makes sense for the solutions to an inequality to be represented by a set of points on one side of a particular value on the number line
Chapter 30
He plans to buy some prepared dishes from a supermarket The prepared dishes are sold by the pound, at $5.29 per pound You also used the Multiplication and Division Properties of Inequality to solve inequalities. When using the Multiplication and Division Properties of Inequality, you must reverse the symbol when you multiply or divide by a negative number
Chapter 31
The boundary between the two regions is the graph of an equation that is related to the inequality You also recalled that a linear equation in two variables has many solutions, which are represented by all the points on the graph of the equation Decide if the values in each ordered pair, ( x , y ) , make the value of the expression less than, greater than, or equal to 12
Chapter 32
This prepared you to write linear inequalities that represent constraints in situations and graphing the solution regions You graphed a related equation, interpreted the coordinate pairs of points on the graph and on either side of the graph, and tested the pairs of values to see if they make the inequality true. You then used these observations to determine the solution region to the inequality You wrote an inequality that represents the constraints in a situation, graphed its solutions, and interpreted points in the solution region
Chapter 33
Connect the different representations and interpret them in terms of the situation, when given inequalities, graphs, and descriptions that represent the constraints in a situation You practiced adjusting the graphing window until the solution regions become visible and give useful information You wrote inequalities in two variables to represent constraints in situations, used technology to graph the solutions, interpreted points in the solution regions, and used the inequalities and the graphs to answer… This sorting and matching task gave you opportunities to analyze representations, statements, and structures closely and to make connections
Chapter 34
You recalled that a solution to a linear equation in two variables is any pair of numbers that makes the equation true, and that a solution to a system of two equations in two variables is a pair of numbers that makes… This served to motivate a desire to represent both constraints on the same graph Your focus was on thinking about each pair of constraints as a system, finding the solution region of each inequality, and identifying a point in the region where the graphs overlap as a solution to the system You began to graph systems of inequalities
Chapter 35
Making comparisons prompted you to think about the solutions to the equations, inequalities, or systems that were represented. Compare each of the following graphs, then list a reason for each graph why it would not belong with the other three Your answers may vary,, but here are some samples
Chapter 36
Step 1 - Examine the following nutrition information for some trail mix ingredients
Chapter 37
A scatter plot is a graph of numerical data with two variables. When the points of the scatter plot seem to fall along a line, the scatter plot is linear If the scatter plot shows no pattern, it means there is no relationship between the variables Distinguish Patterns: Mini-Lesson Review
Chapter 38
Draw a linear model that fits the data well and use the linear model to estimate values You noticed and wondered many things about the images, the relationship between the number of people and the maximum noise level, interpreting the line of best fit, and general information about scatter plots The linear model was also used to interpolate and extrapolate information about the data in context You were given scatter plots for different pairs of variables and the equation of a line of best fit.
Chapter 39
The solid line is the better fit for the data since it goes through the middle of the data and follows the negative trend of the data with very few points a far distance from the line of best fit The solid line is the better fit for the data since it goes through the middle of the data with approximately equal numbers of data points on either side of the line The dashed line is the better fit for the data since it goes through the middle of the data with a slope that follows the trend of the data. The solid line has a lot of points above the line to start and a lot of points below the line at the end
Chapter 40
A residual is the difference between the actual y -value for a point and the expected y -value for the point on the linear model with the same associated x -value A plot of the residuals for data that are fit well by a linear model shows residuals that are close to the x -axis and do not show a noticeable pattern Mentally calculate how close the estimate is to the actual value using the difference
Chapter 41
A doesn’t belong because it is the only scatter plot with data that fit a line well and are increasing B doesn’t belong because it is the only scatter plot with data that would not fit a line well and do not show a pattern Match the correlation coefficient with a scatter plot and linear model During this activity, you had an opportunity to use language precisely and to hear how you use terminology and talk about characteristics of items in comparison to one another
Chapter 42
You thought about whether there would be a strong correlation or not as well as whether the relationship was a positive or negative correlation In addition, you interpreted the coefficients of the equation of the line of best fit and used the equation to make predictions You examined a pair of variables and a correlation coefficient to describe the relationship between the variables as strong or weak and as positive or negative
Chapter 43
You began to recognize that some variables may be related, but one does not always cause the other to change. You were able to recognize that at this point, the mathematics of scatter plot analysis cannot determine whether there is a causal relationship. The relationship must be thought through carefully to decide, based on the situation, whether the related variables have a causal relationship
Chapter 44
Collect data, create a linear model to fit the data, determine if the linear model is a good fit, and use the information from my linear model to answer questions An anthropologist finds a fossilized humerus bone of an ancient human ancestor. The humerus is an arm bone running from the shoulder to the elbow. In this culminating project, you will integrate ideas from the unit as you collect, summarize, interpret, and draw conclusions from bivariate data using scatter plots, best fit lines, and correlation coefficients
Chapter 45
Access the interactive simulator to begin this project Today you are going to investigate something called an interactive simulation. It is a tool that will help you visualize how functions work Step 1 - Open the interactive simulator, Function Builder in your web browser.
Chapter 46
The slope of a line is the ratio of the change in y and the change in x . You can use the slope formula, m = y 2 - y 1 x 2 - x 1 , to find the slope of the line through points ( x 1 , y 1 ) and ( x 2 , y 2 ) Find the slope of the line that goes through points Here’s how to find the slope given two points
Chapter 47
A customer at a bagel shop is buying 13 bagels. You also determined which variable was a function of the other to make statements about the relationship The store worker says, “That would be $16.25.” Jaime, Pedro, and Hannah, who are in the shop, all think it is a mistake
Chapter 48
Each graph represents the distance of a dog from a post as a function of time since the dog owner left to purchase something from a store. After 80 seconds, the dog is 4 feet away from the post You identified the inputs for the functions as they changed You determined if the inputs and outputs created a function
Chapter 49
The output value at f(4) is greater than the output value at f(0) because at 4 seconds the drone is 4 meters off the ground while at 0 seconds, the drone is on the ground (0 meters) The output value at f(2) is equal to the output value at f(5) because at both 2 seconds and 5 seconds, the drone is 4 meters off the ground Here is a graph that represents function f , which gives the height of a drone, in meters, t seconds after it leaves the ground Decide which function value is greater: f ( 0 ) or f ( 4 )
Chapter 50
You also compared output values when the same input was entered into different equations You also determined the meaning of values given function notation in the context of the problem. You then sketched a graph of the function You then used that rule to find other values of the function
Chapter 51
Be able to identify if a graph represents a function through the vertical line test You explained the function notation in words and graphed the functions You also worked backward to find the input value given an output value of a function You explained the meaning of function notation in the given context
Chapter 52
You completed mathematical statements and found the value of variables by reading the information from a graph You identified key features such as minimum and maximum, x -intercepts and y -intercepts, and what they mean in a real-life situation You interpreted statements about a real-life graph and connected them to the function notation that represented them
Chapter 53
Recall that the slope of a line is the rate of change of the linear function. On the graph, it is the steepness of the line Find the slope of the line that connects ( 0 , 0 ) and ( 3 , 2 )
Chapter 54
Estimate or calculate the average rate of change between two points from the graph of a function You made generalizations about the change over time, without finding the specific rate of change You were then able to apply the formula to the generalizations you made in activity 4.8.1 to be more precise in your analysis Once you found the average rate of change, you interpreted the average rate of change in the context of the problem
Chapter 55
Vocabulary words from prior lessons, such as “maximum,” “minimum,” “increasing,“ and “decreasing,” were used to review and to describe the graphs You determined which graph best matched the scenario, and which graphs were the least logical considering the scenario. You also reviewed the graph of a vertical line, including how it is not a function and why it would not work in the real-world scenario You had to share and discuss vocabulary, including the different key features of your graph.
Chapter 56
Make sense of an equation of the form f ( x ) = g ( x ) in terms of a situation and a graph, and know how to find the solutions You had to estimate the values of functions and determine where the graphs were the same in the context of the problem You also used function notation to do calculations It was important to notice and analyze the units of each axis before you could compare the three different graphs.
Chapter 57
The constant term is different in each equation The number that is added to the equation is different You looked at how different slopes and y -intercepts changed the graphs Use the Desmos graphing tool or technology outside the course.
Chapter 58
You noticed the reasonableness of an input depends on the situation’s context You also identified input-output pairs from graphs Earlier, you saw a situation where the total number of times a dog has barked was a function of the time, in seconds, after its owner tied its leash to a post and left.
Chapter 59
You then determined the domain and range when you were given more information about the graphs You also determined the value of the x - and y -intercepts You then determined the domain and range of each type of function. For continuous functions, you wrote the domain and range as inequalities
Chapter 60
You then thought of another rule for the sequence and gave a different list of the next 3 terms Here is a rule for making a list of numbers: Each number is 1 less than twice the previous number Then follow the rule to build a list of 5 numbers In the Tower of Hanoi puzzle, a set of discs sits on a peg, while there are 2 other empty pegs
Chapter 61
The first two sequences are increasing, and the second two are decreasing Geometric sequences can get larger or smaller You used a common ratio to find the pattern to complete the sequence Examine the patterns of numbers given below
Chapter 62
Consider the function f given by f ( n ) = 3 n - 7 . This function takes an input, multiplies it by 3, and then subtracts 7 For each sequence, describe a way to produce a new term from the previous term
Chapter 63
An arithmetic sequence is a sequence in which each term is the previous term plus a constant A geometric sequence is a sequence in which each term is a constant times the previous term The recursive definition is a function that provides a repeated, or recurring, process to find terms in a sequence
Chapter 64
It is possible to use them interchangeably It is the only expression with repeated addition
Chapter 65
For questions 1 - 2 , use the following prompt and image Write an equation for a model that fits the data in the image and gives the percent of battery power as a function of time since the phone was fully charged. For questions 3 - 5, use the following prompt and images If you get stuck, consider creating a table of values or a scatter plot of the data
Chapter 66
You can also do the reverse—identify the slope and y -intercept and use them to find the equation of the line The y -intercept is ( 0 , - 4 ) , and the graph passes through ( 3 , - 2 ) Write Linear Functions from a Graph: Mini-Lesson Review You can determine the slope and intercept of a line if the equation is written in slope-intercept form, y = m x + b .
Chapter 67
To simplify, the exponential terms are added and the same base is kept. You also learned how to use the Quotient Property for Exponents to rewrite division problems involving exponential terms with the same base. To simplify those, the exponential terms are subtracted and the same base is kept To simplify an expression with a negative exponent, the reciprocal of the base is used and the sign of the exponent is flipped
Chapter 68
Notice that the root and exponents are the same and simplify to leave the integer only Rewrite expressions with rational exponents as roots You looked at which operations were equivalent and which ones were inverses of each other You were also introduced to new vocabulary such as radical, index, and radicand and discovered how they are related to perfect squares and square roots
Chapter 69
Begin making connections between linear and exponential growth and the real world (especially financially) You did this by making sense of the provided graphs and used them to answer questions in the context of the scenario. You recalled that a quantity that grows by adding the same amount at each step forms a line when plotted. Also, after viewing an example, you learned that a quantity that grows by the same ratio or factor at each step forms a curve when plotted
Chapter 70
You looked at patterns in exponents to determine the value of zero as an exponent and applied the basic rules of exponents with zero During this activity, you expanded upon the table that was created in 5.4.3 and represented it in a graph Find the value of x that makes the equation true Complete the table by entering the values for a, b, and c below the table.
Chapter 71
Know the meanings of “exponential growth” and “exponential decay.” You also extended the pattern of the equation of each You also learned how to use the values from the problem to write the equation The values of the equations were analyzed in the context of the situation, and each of the observations was graphed
Chapter 72
You used the Product Property of Exponents to simplify the expressions You interpreted the meaning of negative exponents in this equation. You used negative exponents in exponential growth equations to solve real-world problems You also created a graph to represent the situation.
Chapter 73
The numbers were written in both decimal and fraction form, and you may have noticed different patterns in each format You were able to analyze each cell phone and compare them to one another by looking at key features of the graphs Each description was a real-life scenario, and all of the graphs were exponential. You had to consider words that represent different growth rates and whether the situation was growth or decay
Chapter 74
Interpreting the graph in terms of the context helped prepare you for your work with functions in the rest of the unit Here is a graph of the accumulated rainfall in Las Vegas, Nevada, in the first 60 days of 2017 Use the graph to support your answers to the following questions
Chapter 75
Lin and Diego are discussing two expressions: x 2 and 2 x Lin says, “I think the two expressions are equivalent.” Diego says, “I think the two expressions are only equal for some values of x .”
Chapter 76
In particular, the focus for this activity was on how exponential functions have different rates of change for different input intervals, which is not the case for linear functions This time, you focused on a previously encountered exponential decay context Let p be the function that gives the cost p ( t ) , in dollars, of producing 1 watt of solar energy t years after 1977.
Chapter 77
You became more comfortable with the fact that the data are not exactly exponential and with choosing different ways for deciding on an appropriate exponential decay factor To do so, you determined the growth factor of successive bounce heights You processed your data and decided on an appropriate factor to quantify the bounciness of each ball. You used the given data to calculate a rebound factor and used it to write a function that models the relationship between number of bounces and bounce heights.
Chapter 78
She continues spending a third of what is left each week thereafter The equation that results is g = 180 · ( 23 ) t In the first week, she spent a third of the gift money. Then, she spends a third 1 3 of the whole amount each week.
Chapter 79
F is the only one that doubles when n increases by 1 G is the only one that does not have 8 for its vertical intercept H is the only one that is not an exponential function J is the only one that is not increasing
Chapter 80
You also had to justify your response and decide what information would make it easier to determine if the graph was linear or exponential The graph is for y = 120 + 3.7 x because it appears linear. On the graph, when x = 7 , the value of y appears to be a little less than 150.
Chapter 81
For each given expression, write an equivalent expression with as few terms as possible [ 4 ( n + 1 ) + 10 ] - 4 ( n + 1 ) [ 4 ( n + 1 ) + 10 ] - 4 ( n + 1 )
Chapter 82
Use the following prompt and table to answer questions 1 - 7 Write one to two sentences for each city. For each population that you think can be modeled by a linear and/or exponential function: Write an equation for the function(s)
Chapter 83
The exponent 4 means there are 4 factors of y The exponent 2 means there are 2 factors of y Simplify Expressions Using Properties of Exponents: Mini-Lesson Review Notice the exponent is the sum of the exponents of the factors
Chapter 84
Access the interactive simulator to begin this project If you have different responses, try to come to a consensus Today you are going to investigate something called an interactive simulation. It is a tool that will help you visualize how area model multiplication works
Chapter 85
Monomials, binomials, and trinomials are all different types of polynomials. A polynomial is a monomial or two or more monomials combined by addition or subtraction You also found the degree of different polynomials
Chapter 86
Multiplying a polynomial by a monomial is done by applying the Distributive Property of Multiplication over Addition You used properties of exponents when multiplying variables with different exponents You also learned how to multiply conjugate pairs using the Difference of Squares Pattern
Chapter 87
Divide polynomials using synthetic division You also used distributed division to divide polynomials by monomials You also learned how to interpret a remainder in these division problems
Chapter 88
This is an important skill used in factoring polynomials later in the lesson Finding the factors of a number, monomial, or polynomial is an important step used in the activities of this lesson Factor 56 into prime numbers to find all of its factors
Chapter 89
You also learned some tricks to help determine the factors of a trinomial, such as when the middle term is negative and the last term is positive, the signs in the binomial factors must both be negative. By finding the factors of the c term and checking to see if they add up to the b term, you are able to find the factors of a trinomial of this form You practiced finding the factors of a trinomial of the form a x 2 + b x + c using a trial and error method By finding two numbers that multiply to a c and add to b , you can factor the trinomial by grouping.
Chapter 90
Identify which of the polynomials is a perfect square trinomial They always follow a pattern of a 2 + 2 a b + b 2 or a 2 - 2 a b + b 2 You can recognize that they will always factor to a form of ( a + b ) 2 or ( a - b ) 2 . In some cases, you first need to factor the GCF out of the existing terms
Chapter 91
You created sample polynomials that would fit into each strategy When given a sample polynomial, you assessed it and determined an appropriate strategy for factoring You combined strategies to completely factor polynomials of many different forms
Chapter 92
Complete the table with the length, width, and area of each rectangle. Use the given information in the table to find the missing length, width, or area of each rectangle.
Chapter 93
The first term in the result is the product of the first terms in each binomial The second and third terms are the product of multiplying the two outer terms and then the two inner terms You can multiply polynomials using the distributive property
Chapter 94
The x -values are 1 , 2 , 3 , 4 , and 5 in all three tables Create drawings, tables, and graphs that represent the area of a garden You realized that the closer the rectangle was to a square, the larger the area
Chapter 95
Tell whether a pattern is growing linearly, exponentially, or quadratically There are 6 rows with 8 small squares, but from this, 2 rows of 3 small squares have been removed
Chapter 96
Know that, in a pattern of shapes, the step number is the input and the number of squares is the output Figure B is a large square with a smaller square removed. Figure C is composed of two large squares with one smaller square added
Chapter 97
The correct order is 9 2 , 10 2 , 2 9 , 2 10 . In each pattern, the number of small squares is a function of the step number, n You then determined if each expression was linear, exponential, or quadratic You then determined which function had a larger value as the x -values increased
Chapter 98
The big thing you learned is that the discovered patterns help determine the function rule The big thing you learned is that the motion of a falling object is commonly modeled with a quadratic function All y -values are even and divisible by 4, 8, and 16
Chapter 99
The big lesson was that adding a quadratic term to a linear function has an effect of “bending” the graph, as the output values are no longer changing at a constant rate Create quadratic functions and graphs that represent a situation Relate the vertex of a graph and the zeros of a function to a situation Know that the domain of a function can depend on the situation it represents
Chapter 100
Choose a domain that makes sense in a revenue situation Model revenue with quadratic functions and graphs Relate the vertex of a graph and the zeros of a function to a revenue situation You modeled and graphed the relationship as a quadratic function that showed the revenue as a function of the price of the download.
Chapter 101
Rewrite quadratic expressions in different forms by using an area diagram or the distributive property Explain why the diagram shows that 6 ( 3 + 4 ) = 6 · 3 + 6 · 4 The entire diagram represents the product 6 ( 3 + 4 ) . The upper half represents 6 · 3 while the lower half represents 6 · 4
Chapter 102
The sum of the partial products are x 2 + - x + 1 or x 2 - 2 x + 1 Rewrite quadratic expressions given in factored form in standard form using either the distributive property or a diagram Know the difference between “factored form” and “standard form.”
Chapter 103
You then made observations about how the graphs and forms are related in helping identify intercepts These characteristics included x - and y -intercepts Here is a graph of the equation y = 8 - 2 x The 8 in y = 8 - 2 x means the graph intersects the y -axis at 8.
Chapter 104
You used this function to complete a table and determine the x -intercepts and vertex of the graph of the function Here is a graph of a function w defined by w ( x ) = ( x + 1.6 ) ( x - 2 ) . Use the graph to answer the following questions
Chapter 105
Proactive people don’t accept the world as it is or wait for direction from others to initiate change. Graphs A, B, and C represent three linear equations Use the following graph to answer questions 1 - 3
Chapter 106
In the right column, one of the factors is just a variable whose coefficient is 1 or -1, and the other factor is either a sum or a difference Match equations given in standard and factored form with their graph Complete each row with an equivalent expression in standard form or factored form - x ( x - 10 ) or x ( - x + 10 ) or x ( 10 - x )
Chapter 107
The height, in inches, of a frog's jump is modeled by the equation h ( t ) = 60 t - 75 t 2 , where the time t , after the frog jumps, is measured in seconds They mean that the frog is on the ground when t = 0 (before jumping) and when t = 0.8 (when the frog lands on the ground) Explain how you know this is the time the frog reached the maximum height The frog reached its maximum height at t = 0.4 , as that is the halfway point of the jump.
Chapter 108
The expressions are in standard form, factored form, and one other form In each set, the expressions that define the output are equivalent. Relate the numbers in the vertex form of a quadratic equation to its graph
Chapter 109
You graphed the vertex and two other points to complete the graph Expressions in different forms can be used to define the same function. Here are three ways to define a function f
Chapter 110
The value of g is always 4 greater than the value of f , no matter which value of x we choose The expression that defines function f is in factored form. You found the equation of the parabola that modeled the path a peanut needed to take to jump over the wall in a simulation
Chapter 111
In this culminating project, you will integrate ideas from the unit by designing a fountain with given criteria. You will use the fact that the shape made by water coming out of a jet or fountain can be modeled with a quadratic equation In the first activity, you will work through a simplified example with one fountain and one jet. Then, you will think about what information is needed to write the equation and create a sketch to draw the path.
Chapter 112
Write Equivalent Expressions: Mini-Lesson Review The standard form of a quadratic expression is a x 2 + b x + c , where a , b , and c are constants, and a is not 0 Write ( x - 3 ) ( x + 5 ) as an equivalent expression in standard form using the distributive property Step 1 - Multiply each term in the first binomial by the second binomial
Chapter 113
A mechanical device is used to launch a potato vertically into the air. The potato is launched from a platform 20 feet above the ground, with an initial vertical velocity of 92 feet per second The function h ( t ) = - 16 t 2 + 92 t + 20 models the height of the potato over the ground, in feet, t seconds after launch You then used the graph to describe the situation
Chapter 114
A strategy such as this might be difficult to use when the solutions are harder to find You set quadratic equations equal to zero when possible to find solutions in context A quadratic equation that has been set equal to zero provides more opportunities for how to solve it. By developing a better understanding of quadratic equations, you also better understand the context of a real-world situation
Chapter 115
Be able to find solutions to quadratic equations by reasoning about the values that make the equation true; explain that quadratic equations may have one or two solutions.
Chapter 116
Determine which of the statements are true if g · h = 0 . You increased your knowledge of these types of problems that you have seen in previous lessons HINT: There may be more than one statement that is true
Chapter 117
This includes not taking the square root of a negative number and accurately determining how many solutions an equation might have Decide whether each statement is true or false
Chapter 118
The top shape is a square measuring 8 in. Rewrite a quadratic expression in standard form when given the quadratic expression in factored form Rewrite a quadratic expression in factored form when given the quadratic expression in the form of a x 2 + b x + c This lesson covered expressions in the factored forms ( x + a ) ( x + b ) and ( x - a ) ( x - b )
Chapter 119
This built a foundation for understanding how to find the factored forms of different quadratic expressions You learned to distinguish the signs of the factors based on the middle, or linear, term in the quadratic expression In the previous lesson, you experienced the role products and sums play in the way that quadratic expressions are converted between standard and factored forms.
Chapter 120
Rewrite a quadratic expression of the form x 2 - c into its factored form You also learned that some binomials of the form x 2 + c do not have a factored form Finding a product mentally can be done by changing the way you look at a multiplication problem
Chapter 121
Rearrange a quadratic equation to be written as expression in factored form = 0 and find the solutions You substituted a value in for x to determine whether the equation was true Then you used the zero product property on the factored form to solve the quadratic equation You used this relationship to make predictions about the quadratic function
Chapter 122
Being able to recognize detailed aspects of these expressions helps provide a better understanding that is useful when solving by factoring Equation A: The only expression that is not in standard form. By using a guess and check method, you were able to solve these expressions You learned to use one method when the leading coefficient was a perfect square and another method when it was not a perfect square
Chapter 123
Write the quadratic function f ( x ) = x 2 - 9 x + 20 in factored form Graph the quadratic function f ( x ) = x 2 - 9 x + 20 Use the Desmos graphing tool or technology outside the course
Chapter 124
Make predictions about a real-world situation using a quadratic data set You looked for a visual pattern in a sample data set Click on the "+" sign and choose to add a table in the Desmos graphing tool.
Chapter 125
The goal of each round is to write the quadratic expression as it has been given to your partner. Relate Zeros to the Factored Form of Quadratic Equations The equation x 2 - 8 x + 20 = 0 is shown in the graph. Convert between Factored Form and Standard Form
Chapter 126
If a n and b n are real numbers, and n ≥ 2 is an integer, then Simplifying Radicals: Mini-Lesson Review Some square roots can be simplified using the product property.
Chapter 127
One method involved taking the square root of both sides of the equation. The other method involved multiplying two binomials and then factoring
Chapter 128
Explain how you know the expressions you selected are perfect squares Select three expressions that are perfect squares The correct answers are: ( x + 5 ) ( 5 + x ) , ( x - 3 ) 2 , x 2 + 8 x + 16
Chapter 129
Completing the square can be used to solve quadratic equations To complete the square to solve an equation, follow these steps
Chapter 130
Complete the square for quadratic expressions of the form a x 2 + b x + c when a is not 1 and explain the process If it was not a perfect square, you made a suggestion that could make the expression a perfect square Elena says, " ( x + 3 ) 2 can be expanded into x 2 + 6 x + 9 .
Chapter 131
You graphed the equations to find the approximate solutions using a graphing calculator Here are some squares whose vertices are on a grid Find the area and the side length of each square
Chapter 132
Evaluate the expressions and find the two numbers Write down your answer for how to solve each equation.
Chapter 133
The correct answers are 2 and 8: - ( - 5 ) + 9 = 5 + 3 = 8 and - ( - 5 ) - 9 = 5 - 3 = 2 With each one, you identified the error that occurred and explained why it was an incorrect step Evaluate each expression for a = 9 , b = - 5 , and c = - 2
Chapter 134
One way to solve the quadratic equation x 2 + 5 x + 3 = 0 is by completing the square . A partially solved equation is shown here. This helps us avoid some tricky fractions.
Chapter 135
Determine if the vertex for the quadratic function is a maximum or minimum A ball is thrown upward from the top of a 40 foot high building at a speed of 80 feet per second. Write the quadratic function f ( x ) = x 2 + 6 x + 13 in vertex form
Chapter 136
When given a quadratic expression in factored form, rewrite it in standard form When given a quadratic expression in standard form, rewrite it in vertex form These expressions each define the same function
Chapter 137
F ( 1 ) can be expressed in words as "the value of f when x is 1." Find or compute Then you practiced interpreting the language related to maximum and minimum values of functions One was graphed, and the other was in factored form. You used these two pieces of information, without creating a graph of the function in factored form, to determine the maximum value of each one
Chapter 138
A Linear Function and a Quadratic Function Here are graphs of a linear function and a quadratic function. The quadratic function is defined by the expression The function h , defined by h ( t ) = - 5 t 2 + 10 t + 7.5 , models the height of a diver above the water (in meters) t seconds after the diver leaves the board.
Chapter 139
“Failure” gets an undeserved bad reputation in education. Research stretching back to the earliest days of cognitive psychology suggests that we may learn more from failure than from success 9 . However, many educational practices, such as grading, intrinsically punish and stigmatize failure 5 instead of celebrating it for what it is: an opportunity for deeper understanding Converging evidence suggests that letting students struggle to figure out a problem before starting direct instruction actually boosts their conceptual understanding of the topic 1 , 2 , 3 , 4 , 6 , 9 , 10 .
Chapter 140
Meaningful learning that can transfer to new contexts is more likely to occur in classrooms with rich classroom discourse. Classroom discussion is a form of generative learning. It is facilitated by teachers who give thoughtful cues and purposeful questions to get conversations flowing and keep them on track Rich classroom discourse is characterized by a handful of features (Hattie, Fisher, & Frey, 2017)
Chapter 141
There is a scourge of what can be called “false growth mindsets” among educators. An example of this would be teachers who claim to endorse a growth mindset and use some of the language of growth mindsets (e.g. Praising effort) while still implicitly endorsing fixed mindset ideals (e.g. “you tried your best,” implying that their “best” is limited and immobile)
Chapter 142
Spaced retrieval practice is the process of distributing many, short sessions of learning over a longer period of time, as opposed to cramming fewer, longer sessions of learning over a shorter period of time. Spaced practice can promote long-lasting memory of targeted information. Spacing study out into multiple, shorter sessions promotes greater long-term learning than cramming study into fewer, longer study sessions.
Chapter 143
That said, it turns out the notion of “learning styles” is not actually supported by research, and in fact a large body of work suggests that everybody learns better through a combination of words and pictures. As educators, we have all heard about or learned about “learning styles” such as visual learners vs auditory learners. This body of work has culminated in the “Multimedia Learning Principle” which asserts that people learn better from words and pictures than from words alone (Mayer, 2022) The theory behind the Multimedia Learning Principle is related to “cognitive load.” It appears humans have separate working memory loads for our auditory and visual channels.
Chapter 144
Students are asked to consider what happens when a pattern is generated from a specific value that is repeatedly added or subtracted (linear). Then, they engage in thinking about patterns generated from a specific value that is repeatedly multiplied or divided (exponential). Then, in later units, students are asked to consider patterns that adhere to a model that both increases and decreases over time (quadratic). Across the nine units of this course, students progress from linear to exponential to quadratic functions, by exploring what makes the functions different.
Before the desk
Margins: A study assistant with links to OpenStax resources
The books on this site are CC BY 4.0 with OpenStax.org, Rice University. They are not affiliated with this platform.
Margins is a study desk beside the OpenStax books, for a first college course, a return to one, or study on your own. The work is grouped by subject: a lesson on what each chapter asks you to know, and practice that asks it back. Scores stay in this browser.
Here, you can find detail on the Resources.
© 2026 PZPT TYRIS Collective LLC. All rights reserved. Branford. Tyris.
The selection and arrangement of this site are claimed by PZPT TYRIS Collective LLC. Material produced by Grok is not claimed except where a person wrote or rewrote it.
Created through Grok AI Build.