# Linear Function Questions on the SAT

Many SAT students have asked me: “If you had just one hour to prepare for the SAT, what would you study?” My answer is absolute and unwavering: “I would spend 50 minutes of that hour on linear functions.” If you’re wondering why, it’s because more SAT math questions involve lines, linear equations, linear growth and decay, and linear systems than any other concept.

In this article, we will present you with many linear function questions that you may encounter on the SAT quant section. However, the topic is so broad that we can’t cover every example of linear functions in one article. You can find comprehensive coverage of the topic in our SAT self-study course. From the basics (y = mx + b) to the sophisticated (scatterplot regression lines), you’ll find “all things linear” at our course.

## What Is a Linear Equation?

There are three criteria that an equation needs to meet in order to qualify as a linear equation:

• It contains one or more variables
• Each variable is raised to the first power
• None of the variables are multiplied together

Here are some examples of linear equations:

• 2x = 4 – x  (a single-variable linear equation)
• y = 3x – 8 (a two-variable linear equation)
• x + y = 5 – 4z (a multi-variable linear equation)

The following are not linear equations:

• y = 2x^2 – 6x – 20 (the variable x is squared)
• y = 5 / x → xy = 5 (the variables are multiplied together)

KEY FACT:

A linear equation contains one or more variables, each of which is raised to the first power, and none of which are multiplied together.

Let’s try an example.

### Example 1: Identifying a Linear Equation

Which of the following represents a linear equation?

• 3x + 4^2 = -2y
• 6 – y = 3x^2
• xy = 3

• I only
• II only
• III only
• I and III only

#### Solution:

Let’s consider each of the numbered statements.

Statement I: The equation contains two variables, and they are both raised to the first power. The variables are not multiplied together. Thus, 3x + 4^2 = -2y is a linear equation.

Statement II: The equation contains two variables, but one of them is raised to the second power. Thus, 6 – y = 3x^2 is not a linear equation.

Statement III: The equation contains two variables, but they are multiplied together. Thus, xy = 3 is not a linear equation.

## What Is a Linear Function?

A linear function is defined as a function whose graph is a straight line in the coordinate plane. We know that y = 4x – 8 is a line expressed in slope-intercept form, and so it is a linear function.  We can replace y with the function notation f(x) and re-express the equation of the line as f(x) = 4x – 8. Note that generally we can use “y” and “f(x)” interchangeably. For our purposes in SAT preparation, we will adopt this convention.

For the linear function y = mx + b, or , equivalently, f(x) = mx + b, we review the following from algebra class:

• m = the slope of the line
• b = the y-intercept of the line
• x = the independent variable
• y  or f(x) = the dependent variable

KEY FACT:

A linear function’s graph is a straight line in the coordinate plane, and it is generally expressed as f(x) = mx + b.

## Dissecting the Slope-intercept Form of a Line

The two key players in the slope-intercept equation of a linear function f(x) = mx + b are the slope m and the y-intercept b. Let’s review what they are and how to calculate them, using the following mini-example:

What is the equation of the line that passes through the points (2, 6) and (-1, 2)?

Let’s first calculate the value of the line’s slope, using the formula

Slope = (y2 – y1) / (x2 – x1).

The first ordered pair tells us that x1 = 2 and y1 = 6, and the second ordered pair tells us that x2 = -1 and y2 = 2. Substituting these values into the formula gives us:

Slope = (2 – 6) / (-1 – 2) = -4 / -3 = 4/3

At this point, we can write our linear function as:

f(x) = (4/3)x + b

KEY FACT:

We calculate the slope of a line with the formula (y2 – y1) / (x2 – x1), where (x1, y1) and (x2, y2) are ordered pairs through which the line passes.

To determine the y-intercept, we can substitute the x- and y-values from either of the ordered pairs that we just used to determine the slope. Let’s use (2, 6):

6 = (4/3)(2) + b

6 = 8/3 + b

6 – 8/3 = b

18/3 – 8/3 = b

10/3 = b

Our y-intercept is 10/3. Thus, we can express the equation of the linear function as:

f(x) = (4/3)x + 10/3

### Example 2:  Determining the Equation of a Line

What is the equation of the line passing through the points (1, 2) and (5, -6)?

• f(x) = -2x – 4
• f(x) = -2x + 4
• f(x) = -2x
• f(x) = 2x + 4

#### Solution:

Let’s first calculate the value of the line’s slope, using the formula

Slope = (y2 – y1) / (x2 – x1).

The first ordered pair (1, 2) tells us that x1 = 1 and y1 = 2, and the second ordered pair (5, -6) tells us that x2 = 5 and y2 = -6. Substituting these values into the formula gives us:

Slope = (-6 – 2) / (5 – 1) = -8 / 4 = -2

At this point, we can write our linear function as:

f(x) = (-2)x + b

To determine the y-intercept, we can substitute the x- and f(x) values from either of the ordered pairs that we just used to determine the slope. Let’s use (1, 2):

2 = (-2)(1) + b

2 = -2 + b

4 = b

Our y-intercept is 4. Thus, we can express the equation of the linear function as:

f(x) = -2x + 4

## Linear Function Word Problems

You will most likely encounter a basic linear function word problem on the SAT. One of the more common types is one we call “Rental Problems.” In these types of problems, you rent an item, paying a fixed base price and a variable additional price, based on the amount of time you used the item. This situation is modeled by a linear function.

Let’s look at a mini-example.

You rent a car for $80 for the week. Additionally, you pay$0.30 for every mile you drive the car. At the end of the week, you have traveled 560 miles. How much is your total rental fee for the week?

First, we know that the fixed cost is $80. To determine the variable cost, let’s let x = the number of miles traveled during the week. Since we pay$0.30 for each mile, the total variable cost is 0.30x. We can set up an equation for this real-world situation, as follows:

Total Cost = Fixed Cost + Variable Cost

Total Cost = 80 + 0.30x

Before we solve the problem, note that the equation is actually linear, where 80 (the constant) is the y-intercept and 0.30 is the slope. Let’s re-express the equation into our familiar f(x) = mx + b form:

f(x) = 0.30x + 80

We obtain the total cost of the week’s rental by substituting the number of miles traveled, 560, for x:

f(560) = (0.30)(560) + 80

f(560) = 168 + 80

f(560) = 248

Thus, the total cost of the rental is $248 when the car has been driven for 560 miles. KEY FACT: A “rental problem” describes a real-world context in which we have a fixed cost (b) and a variable cost (m) modeled by the linear function f(x) = mx + b. Let’s do an example question. ### Example 3: Rental Problem: Fixed and Variable Costs Jerome has two options for renting a jackhammer. Option 1: he can rent it for a flat fee of$30 plus $5 for each hour after the second hour. Option 2: he can rent it for$10 per hour.

For what number of hours will the two rental prices be equal?

• 3
• 4
• 5
• 6

#### Solution:

Option 1: The fixed cost is $30. If we let x = the total number of hours that Jerome rented the jackhammer, then, he pays the hourly rate of$5 for (x – 2) hours, since he pays for hours only in excess of 2. Thus, the total cost f(x) for Option 1 is:

f(x) = 30 + 5(x – 2)

Option 2: For this option, there is no fixed cost. The variable cost is $10 per hour, and he rents the jackhammer for x hours. Thus, the total cost for Option 2 is: f(x) = 10x We want to determine the number of hours for which the two costs are equal. Thus, we can set the two linear functions equal to each other and solve for x: 30 + 5(x – 2) = 10x 30 + 5x – 10 = 10x 20 = 5x x = 4 Answer: B ## Interpretation of the Y-Intercept A common question that you might encounter on the SAT is one asking you to interpret the meaning of either the y-intercept or the slope in the context of the problem. Let’s use the previous example to illustrate the meaning of these two concepts. First, let’s consider the interpretation of the y-intercept. You rent a car for$80 for the week. Additionally, you pay $0.30 for every mile you drive the car. At the end of the week, you have traveled 560 miles. How do we interpret the y-intercept in the context of this problem? The y-intercept is the value of the function when x = 0. For this linear function, we determined that the equation of the line was: f(x) = 0.30x + 80 Thus, when x = 0, the value of the function is f(x) = 80. So, we can interpret the y-intercept as the car rental cost if we have traveled 0 miles, which is$80.

KEY FACT:

The y-intercept is the value of the linear function when the variable x = 0.

Let’s consider the following example.

### Example 4: Interpretation of the Y-Intercept

A t-shirt company charges a one-time fee of $40 plus$5.00 per t-shirt for printing a logo on the t-shirt. The equation for the total cost to print logos on x t-shirts is:

f(x) = 40 + 5x

What is the value and interpretation of the y-intercept in the context of this question?

• It is the value 5 when x t-shirts are printed with a logo
• It is the value 0 when no t-shirts are printed with a logo
• It is the value 40 when no t-shirts are printed with a logo
• It is the value 40 when x t-shirts are printed with a logo

#### Solution:

The y-intercept is the value of the function when x = 0, in other words, when no t-shirt has had a logo printed on it. We show this algebraically as

f(0) = 40 + 5(0)

f(0) = 40 + 0

f(0) = 40

In other words, when no t-shirts have been printed with a logo, the cost is $40. This is the interpretation of the y-intercept of this linear function. Answer: C ## Interpretation of the Slope Let’s revisit the example about the cost of printing t-shirt logos, which we modeled with the linear function f(x) = 40 + 5x. Note that the y-intercept is the constant 40, and the slope is 5. We discovered that the y-intercept describes the fixed cost of the job, when no t-shirt logos have been printed. Now, let’s look at how we interpret the slope. The slope of 5 tells us that for each additional t-shirt logo printed, the total cost of the job increases by$5. Let’s see how this works by considering the cost of printing 1 t-shirt compared to the cost of printing 2 t-shirts:

f(1) = 40 + 5(1) = $45 f(2) = 40 + 5(2) =$50

Notice that when we calculated the cost of one more t-shirt, the total cost increased by $5. The value of the slope in the cost function equation f(x) = 40 + 5x gives us this information. KEY FACT: The slope of a line indicates the additional amount of the dependent variable (y-value) for a 1-unit increase in the independent variable (x-value). Let’s practice this concept with an example question. ### Example 5: Interpretation of the Slope The number of people at a community swimming pool can be modeled by the linear function f(x) = 30 + 22x, where x is the number of hours after 9:00 a.m. What is the interpretation of the slope in the context of this linear function? (Assume that no one leaves the pool during the day.) • the number of people at the pool at 9:00 a.m. • the number of people at the pool at the end of the day • the number of people entering the pool each hour • The number of hours it takes to reach the pool’s capacity #### Solution: We see that the y-intercept is 30, and the slope is 22. The y-intercept tells us the initial number of people at the pool, which is 30. Since the variable x is expressed in hours, the slope of 22 tells us that 22 people enter the pool each hour. Answer: C ## Linear Growth and Linear Decay Another concept commonly tested on the SAT is linear growth or linear decay. These terms refer to whether an entity is increasing in value or decreasing in value. For example, if a worker charges a trip fee of$15 and an hourly rate of \$25, then as the number of hours increases, the total cost of the job increases. The equation of this example is f(x) = 15 + 25x, where x = the number of hours worked. This is an example of linear growth. Note that the slope of the line is positive.

In contrast, linear decay occurs when the total decreases. For example, let’s say the number of cookies for sale at a bakery is modeled by the linear function f(x) = 220 – 24x, where x is the number of hours after the bakery opens at 6:00 a.m. We see that the y-intercept 220 is the initial number of cookies for sale at 6:00 a.m. But after 1 hour, the total number of cookies for sale is now f(1) = 220 – 24(1) = 196, and the total number of cookies for sale decreases each hour. Because the total decreases, the situation is described as linear decay. You can identify linear decay by noting that the slope is negative.

KEY FACT:

You will have linear growth when the slope of the linear function is positive. You will have linear decay when the slope of the linear function is negative.

Let’s practice with an example.

### Example 6: Linear Growth and Decay

Which of the following equations illustrates linear decay?

• f(x) = 45x – 10
• f(x) = -3x^2 + 7
• f(x) = 5 – (1/2)x

• I only
• II only
• III only
• I, II, and III

#### Solution:

Let’s consider each option individually.

Option I: This is a linear equation, but the slope of 45 is positive. Thus, this is an example of linear growth, not decay. Eliminate choice A.

Option II: This is a quadratic equation, not linear. Thus, it does not illustrate linear decay. Eliminate choice B.

Option III: This is a linear equation, and the slope of -1/2 is negative. Thus, this is an example of linear decay.

## Summary

We have focused on “all things linear” in this article. We learned that a linear function can have one or more variables, each of which is raised only to the first power and none of which are multiplied together. We saw that the linear function can be expressed as f(x) = mx + b, where m is the slope and b is the y-intercept (the y-value when x = 0)

When a real-world linear example is modeled, we interpret the y-intercept as the initial amount or cost. The slope is the additional amount or cost for a 1-unit change in time or other independent variable.

Finally, linear growth and decay questions are common on the SAT. The key to remember is that a linear function with a positive slope involves linear growth, and a linear function with a negative slope indicates linear decay.