Linear Functions

Linear Functions

SAT Math | Algebra | Foundation to Advanced

A linear function describes a relationship between two quantities that changes at a constant rate. On the SAT, you must understand how linear functions work, evaluate their outputs, determine slopes and intercepts, interpret graphs and tables, and create equations from real-world situations.

1. Introduction to Linear Functions

A function is a rule that assigns exactly one output to each input in its domain.

A linear function is a function with a constant rate of change. Its graph is a straight line.

A linear function can be written in the form:

f(x) = mx + b

Where:

• x is the input or independent variable.

• f(x) is the output or dependent variable.

• m is the slope or rate of change.

• b is the y-intercept or initial value.

For example:

f(x) = 3x + 5

Here, the slope is 3 and the y-intercept is 5.

This means that whenever x increases by 1, the output increases by 3.

Consider the following values:

Notice that the output increases by exactly 3 whenever the input increases by 1.

This constant change is what makes the function linear.

Important: The y-intercept is not always positive, and the slope can be positive, negative, or zero.

On the SAT, linear functions may appear as equations, tables, graphs, written descriptions, or combinations of these representations.

2. Function Notation and Input-Output Values

Function notation describes the output associated with a particular input.

For example:

f(x) = 4x − 7

The expression f(3) means the output when x = 3.

It does NOT mean f multiplied by 3.

To find f(3), substitute 3 for x:

f(3) = 4(3) − 7

f(3) = 12 − 7

f(3) = 5

Therefore:

f(3) = 5

The corresponding input-output pair is (3, 5).

Worked Example 1 — Evaluating a function

Difficulty: Easy

The function f is defined by:

f(x) = 6x + 4

What is f(5)?

Step 1: Replace x with 5.

f(5) = 6(5) + 4

Step 2: Multiply.

f(5) = 30 + 4

Step 3: Add.

f(5) = 34

Final answer: 34

Worked Example 2 — Finding the input

Difficulty: Medium

The function g is defined by:

g(x) = 5x − 8

For what value of x is g(x) = 27?

Step 1: Set the function equal to 27.

5x − 8 = 27

Step 2: Add 8 to both sides.

5x = 35

Step 3: Divide by 5.

x = 7

Final answer: 7

SAT tip: Sometimes the output is given and you must find the input. Check carefully whether the question asks for x or f(x).

3. Understanding Slope and Rate of Change

The slope describes how quickly a function's output changes as its input changes.

For two points (x₁, y₁) and (x₂, y₂):

Slope = (y₂ − y₁) / (x₂ − x₁)

The slope is commonly represented by m.

It is also called:

• Rate of change

• Change in output per unit change in input

• Rise divided by run

Positive slope

If m is positive, the function increases as x increases.

Example:

f(x) = 4x + 3

For every increase of 1 in x, the output increases by 4.

Negative slope

If m is negative, the function decreases as x increases.

Example:

f(x) = −2x + 10

For every increase of 1 in x, the output decreases by 2.

Zero slope

If m = 0, the output remains constant.

Example:

f(x) = 7

Every input produces an output of 7.

Worked Example 3 — Finding slope from two points

Difficulty: Medium

A linear function passes through the points:

(2, 7) and (6, 19)

What is its rate of change?

Step 1: Write the slope formula.

m = (y₂ − y₁) / (x₂ − x₁)

Step 2: Substitute the coordinates.

m = (19 − 7) / (6 − 2)

Step 3: Simplify.

m = 12 / 4

m = 3

Final answer: 3

SAT tip: Be consistent with the subtraction order. If you subtract the first output from the second, you must subtract the first input from the second as well.

4. The Y-Intercept and Initial Value

A linear function is commonly written as:

f(x) = mx + b

The number b represents the output when x = 0.

Therefore:

f(0) = b

The point where the graph crosses the y-axis is called the y-intercept.

For example:

f(x) = 5x + 12

Substitute x = 0:

f(0) = 5(0) + 12

f(0) = 12

Therefore, the y-intercept is 12, and the graph crosses the y-axis at (0, 12).

Interpreting the y-intercept

In real-world SAT questions, the y-intercept often represents:

• An initial quantity

• A starting population

• A fixed fee

• An original measurement

• An amount present at time zero

For example:

C(h) = 15h + 40

If C(h) is the total cost of a service after h hours:

• 15 represents the hourly charge.

• 40 represents the fixed starting fee.

Even when no hours are used, the model gives a starting cost of $40.

Important: In contextual questions, the y-intercept only has a meaningful real-world interpretation if an input of zero makes sense for the situation.

5. Writing a Linear Function

When you know the slope and y-intercept, use:

f(x) = mx + b

Worked Example 4 — Slope and intercept

Difficulty: Easy

A linear function has a slope of 4 and a y-intercept of −3.

Which equation defines the function?

Step 1: Use the general form.

f(x) = mx + b

Step 2: Substitute m = 4 and b = −3.

f(x) = 4x − 3

Final answer: f(x) = 4x − 3

Writing a function from two points

If the slope and y-intercept are not directly provided, first find the slope using the two given points.

Then substitute one of the points into f(x) = mx + b to find b.

Worked Example 5 — Function from two points

Difficulty: Medium

A linear function satisfies:

f(2) = 7

f(6) = 19

Which equation defines f?

Step 1: Write the corresponding points.

(2, 7) and (6, 19)

Step 2: Find the slope.

m = (19 − 7) / (6 − 2)

m = 12 / 4

m = 3

Step 3: Write the function in slope-intercept form.

f(x) = 3x + b

Step 4: Substitute the point (2, 7).

7 = 3(2) + b

7 = 6 + b

b = 1

Step 5: Write the final equation.

f(x) = 3x + 1

Check:

f(2) = 3(2) + 1 = 7

f(6) = 3(6) + 1 = 19

Final answer: f(x) = 3x + 1

6. Linear Functions Represented by Tables

The SAT may give a table rather than an equation.

To determine the rate of change, compare the change in output with the corresponding change in input.

Consider:

From x = 1 to x = 3:

Change in input = 2

Change in output = 6

Therefore:

Slope = 6 / 2 = 3

Check the next pair:

From x = 3 to x = 5, the input increases by 2 and the output increases by 6 again.

The rate is constant.

Worked Example 6 — Finding a missing output

Difficulty: Medium

The table represents a linear function.

What is f(8)?

Step 1: Find the slope.

m = (16 − 10) / (3 − 1)

m = 6 / 2 = 3

Step 2: Write the function.

f(x) = 3x + b

Step 3: Use the point (1, 10).

10 = 3(1) + b

b = 7

Step 4: Therefore:

f(x) = 3x + 7

Step 5: Substitute x = 8.

f(8) = 3(8) + 7

f(8) = 31

Final answer: 31

SAT tip: If the input increases by 2, 3, or another amount, do not mistake the total output increase for the slope. Divide by the input change.

7. Understanding Linear Function Graphs

The graph of a linear function is a straight line.

A function can be described using:

• Its slope

• Its y-intercept

• Two points on the line

• A table of values

• Its equation

Reading the slope from a graph

Choose two points whose coordinates can be read accurately.

Find:

Slope = Vertical change / Horizontal change

A line rising from left to right has positive slope.

A line falling from left to right has negative slope.

A horizontal line has zero slope.

Reading the y-intercept

Find where the line crosses the vertical y-axis.

At this point, x = 0.

For example, a line crossing the y-axis at (0, 5) has a y-intercept of 5.

Worked Example 7 — Graph interpretation

Difficulty: Medium

A straight line passes through the points (0, 4) and (3, 13).

Which equation represents the function?

Step 1: Find the slope.

m = (13 − 4) / (3 − 0)

m = 9 / 3 = 3

Step 2: Find the y-intercept.

The line passes through (0, 4), so b = 4.

Step 3: Write the equation.

f(x) = 3x + 4

Final answer: f(x) = 3x + 4

SAT tip: If a graph shows the y-intercept clearly, read it directly rather than calculating it unnecessarily.

8. Interpreting Linear Functions in Real-World Situations

About 30% of SAT Math questions are presented in a real-world or scientific context.

For linear functions, a useful model is:

Output = Starting value + Rate × Input

For example:

C(h) = 25 + 12h

where C(h) represents total cost in dollars after h hours.

The number 25 represents the initial fee.

The number 12 represents the cost per hour.

The input h represents the number of hours.

Worked Example 8 — Real-world modeling

Difficulty: Medium

A water tank initially contains 80 liters of water. Water enters the tank at a constant rate of 6 liters per minute.

Which function gives the amount of water W(t), in liters, after t minutes?

Step 1: Identify the starting value.

80 liters

Step 2: Identify the rate of change.

6 liters per minute

Step 3: Write the function.

W(t) = 80 + 6t

Final answer: W(t) = 80 + 6t

Worked Example 9 — Decreasing relationship

Difficulty: Medium

A machine is worth $9,000 and loses $500 in value each year. Assume its value decreases linearly.

Which function represents its value V(t), in dollars, after t years?

Step 1: Identify the initial value.

9,000

Step 2: Identify the rate.

−500 per year

Step 3: Write the function.

V(t) = 9000 − 500t

The negative slope shows that the machine loses value over time.

9. Comparing Two Linear Functions

The SAT may represent one function using an equation and another using a table or graph.

You may need to compare:

• Rate of change

• Initial value

• Output at a particular input

• Whether the functions increase or decrease

Worked Example 10 — Comparing rates

Difficulty: Medium

Function f is defined by:

f(x) = 5x + 2

Function g is represented by the table.

Which function has the greater rate of change?

Step 1: Identify the slope of f.

m = 5

Step 2: Find the slope of g.

m = (14 − 8) / (2 − 0)

m = 6 / 2 = 3

Step 3: Compare.

5 > 3

Final answer: Function f has the greater rate of change.

Important distinction

A function with a greater slope does not necessarily have a greater output for every input.

For example:

f(x) = 5x + 2

g(x) = 3x + 8

Function f has a larger slope, but g(0) = 8 is greater than f(0) = 2.

Always check what the question is comparing.

10. Advanced SAT Linear Function Questions

Harder SAT questions may include unknown constants, algebraic expressions, negative rates, or information that requires several steps.

Worked Example 11 — Unknown slope

Difficulty: Medium–Hard

The linear function f is defined by:

f(x) = kx + 4

If f(5) = 19, what is k?

Step 1: Substitute the given input and output.

19 = 5k + 4

Step 2: Subtract 4.

15 = 5k

Step 3: Divide by 5.

k = 3

Final answer: 3

Worked Example 12 — Finding the initial value

Difficulty: Hard

A linear function satisfies:

f(4) = 13

f(10) = 31

What is f(0)?

Step 1: Find the slope.

m = (31 − 13) / (10 − 4)

m = 18 / 6 = 3

Step 2: Write the function.

f(x) = 3x + b

Step 3: Substitute (4, 13).

13 = 3(4) + b

13 = 12 + b

b = 1

Step 4: Since f(0) = b:

f(0) = 1

Final answer: 1

Worked Example 13 — Using the constant rate directly

Difficulty: Hard

A linear function f has a rate of change of 6.

If f(4) = 31, what is f(10)?

Step 1: Find the change in input.

10 − 4 = 6

Step 2: Multiply by the rate of change.

6 × 6 = 36

Step 3: Add the output change to the known output.

f(10) = 31 + 36

f(10) = 67

Final answer: 67

SAT shortcut: For a linear function, you can find a new output using the rate of change without first writing the entire equation.

11. SAT Strategies and Common Mistakes

Strategy 1 — Identify the slope and intercept

When a function is written as f(x) = mx + b, identify m and b immediately.

Strategy 2 — Check what the question asks

The SAT may ask for:

• f(x)

• x

• m

• b

• A missing table value

• An input-output pair

• A rate or initial amount in context

Do not confuse these quantities.

Strategy 3 — Use the representation that requires the least work

If the y-intercept is clearly shown on a graph, read it directly.

If a table gives two points, calculate the slope from those points.

If an equation is already in slope-intercept form, read the slope directly.

Strategy 4 — Use units

In word problems, a slope may represent dollars per hour, liters per minute, people per month, or another rate.

Include units when interpreting the answer.

Strategy 5 — Look for a constant rate

Linear functions have constant rates of change.

For equal increases in input, their outputs change by equal amounts.

Common Mistake 1 — Confusing slope and intercept

In f(x) = 4x + 9:

Slope = 4

Y-intercept = 9

Do not reverse them.

Common Mistake 2 — Misunderstanding function notation

f(3) is the output when x = 3.

It is not f multiplied by 3.

Common Mistake 3 — Incorrect slope formula

Always divide the change in output by the corresponding change in input.

Do not divide the change in input by the change in output.

Common Mistake 4 — Ignoring negative signs

A negative slope indicates that the function decreases as the input increases.

Common Mistake 5 — Using only the output difference

If the output increases by 12 while the input increases by 4, the slope is 3, not 12.

Common Mistake 6 — Assuming the greater slope always gives the greater output

A function's output depends on both its slope and its initial value.

Common Mistake 7 — Ignoring the context

A slope of −5 may represent a decrease of 5 dollars per month or 5 liters per minute.

Interpret it using the problem's quantities and units.

Common Mistake 8 — Treating every relationship as linear

Check whether the rate of change is constant.

If the slope changes between intervals, the relationship is not represented by one linear function over the whole interval.

12. Quick Knowledge Check

Try these questions before starting the full practice bank.

Question 1

If f(x) = 4x + 6, what is f(5)?

A. 20

B. 24

C. 26

D. 30

Answer: C — 26

f(5) = 4(5) + 6 = 26.

Question 2

A linear function passes through (2, 5) and (6, 17). What is its slope?

A. 2

B. 3

C. 4

D. 6

Answer: B — 3

Slope = (17 − 5) / (6 − 2) = 12 / 4 = 3.

Question 3

A linear function has slope −2 and y-intercept 9. Which equation defines the function?

A. f(x) = 2x + 9

B. f(x) = −2x − 9

C. f(x) = 9x − 2

D. f(x) = −2x + 9

Answer: D — f(x) = −2x + 9

In f(x) = mx + b, substitute m = −2 and b = 9.

Question 4

The function C(t) = 25 + 7t represents a total cost in dollars after t hours.

What does 7 represent?

Answer: The cost increases by $7 for each additional hour.

The coefficient of t represents the hourly rate.

13. Lesson Summary

A linear function has a constant rate of change and produces a straight-line graph.

The standard slope-intercept form is:

f(x) = mx + b

The slope m measures the change in output per unit change in input.

The y-intercept b represents the output when x = 0.

You can evaluate functions by substituting input values.

You can find unknown inputs by setting the function equal to a given output.

You can determine the equation of a linear function from its slope and intercept or from two points.

Tables and graphs can be used to calculate rates of change, identify intercepts, and predict outputs.

Real-world linear functions often describe fixed amounts combined with constant increases or decreases.

When comparing linear functions, always distinguish between slope, initial value, and output.

You are now ready to complete the Linear Functions practice questions in Baloo Prep.