Linear functions

A linear function changes at a constant rate. Most questions come down to finding or interpreting two numbers: the rate of change (slope) and the starting value (y-intercept).

Tested on SAT High frequency · Algebra

What College Board tests

Writing, evaluating, and interpreting functions of the form . You'll find slope from two points or a table, interpret what the slope and intercept mean in a real-world model, and read values from a graph.

The two numbers that define a line

Slope is "per each" (per hour, per item); the intercept is the "even if zero" amount (the flat fee, the starting population).

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Four worked examples in SAT format. Read the approach, try it yourself, then tap Show the full solution.

1 · Slope from two points

A line in the xy-plane passes through the points and . What is the slope of the line?

Approach Slope is the change in divided by the change in . Subtract in the same order top and bottom — pick one point as "first" and stay consistent.

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Answer: B

the slope formula

substitute (1, 3) and (4, 9)

on top; on the bottom

Why the other choices are wrong

A  Flips the formula — change in over change in .

C  Uses only the bottom difference .

D  Uses only the top difference .

2 · Interpret slope and intercept in context

A plumber's total charge, in dollars, for a job lasting hours is given by . Which of the following is the best interpretation of the number 45 in this context?

1. The plumber charges $45 for the job regardless of how long it takes.
2. The plumber charges $45 for each hour worked.
3. The plumber charges $45 as a one-time fee.
4. The job takes 45 hours to complete.

Approach Match each number to its role. The number multiplied by the variable is the slope — the per-hour rate. The number standing alone is the intercept — the fixed amount.

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Answer: B

the model

45 multiplies — it's the rate per hour

30 stands alone — it's the fixed fee (the charge when )

To see why 45 is a per-hour rate, track the units. "$45 per hour" means "$45 for each one hour," so its denominator is 1 hour:

For hours, the "hour" unit cancels, leaving dollars:

So 45 is the number of dollars added for each additional hour worked.

Why the other choices are wrong

A  Describes a fixed amount — that's the 30, not the 45.

C  A one-time fee is the intercept (30), not the hourly rate.

D  Misreads 45 as a duration rather than a rate.

3 · Build a function from a table

The table shows several values of a linear function .

x	0	1	2
f(x)	5	8	11

What is the value of ?

Approach Find the two numbers. The output at is the intercept. The amount the output jumps for each step of 1 in is the slope. Then build and plug in 10.

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Answer: C

read the intercept from the table: when , — the column gives

slope is the change in output over the change in input, from to

check the next pair, to — also 3, confirming the rate is constant

build the function: with ,

substitute 10 for

Why the other choices are wrong

A  Uses a slope of 2 instead of 3 somewhere in the build.

B  Computes but forgets to add the intercept.

D  Adds the intercept twice.

4 · Find where a line crosses an axis

A line in the xy-plane is defined by . At what value of does the line cross the x-axis?

Approach A line crosses the x-axis where . Substitute and solve for .

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Answer: C

the given equation

at the x-axis,

the term becomes 0

divide both sides by 2

Why the other choices are wrong

A  Sign error when isolating .

B  Sets instead of — that finds the y-axis crossing.

D  Reads the constant 20 directly without dividing.

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Quick reference