ORIGINAL WORKED EXAMPLE
Substitute an expression
For f(x) = 4x − 7, simplify f(t + 2) − f(t).
- f(t + 2) = 4(t + 2) − 7 = 4t + 1.
- f(t) = 4t − 7.
- Subtract the complete second expression: (4t + 1) − (4t − 7) = 8.
Answer 8
SAT · Algebra · TOPIC LESSON
Linear functions
Connect constant rate of change with function notation, tables and contextual models.
BY THE END, YOU CAN
- Evaluate and interpret linear functions in several representations.
- Construct a function from a rate and an initial value or two data points.
- Describe input/output changes, zeros and domain restrictions.
BEFORE YOU START
- Slope and solving linear equations.
- Reading a table of ordered pairs.
Original educational lesson. Work through the examples before revealing the practice answers. This lesson supports SAT Math and Reading and Writing skill study; it does not reproduce official questions or guarantee a score.
LEARN THROUGH ORIGINAL PRACTICE
Original SAT-style worked examples and practice
The teaching passages, worked examples and exercises in this lesson are original. They are not official College Board questions or past-exam reproductions. Blueprint and observed-practice analytics describe separate factual metadata; those source questions are not displayed here.
01 / UNDERSTAND & APPLY
A function assigns one output to each allowed input
A function is a rule that gives exactly one output for each input in its domain. The notation f(3) means the output when the input is 3, not f multiplied by 3. A linear function has the form f(x) = mx + b; its graph is a nonvertical line. A vertical line is not a function of x because the same x would have multiple outputs.
The number m is the constant rate of change and b = f(0) is the output at zero input. A constant function, m = 0, is included here.
02 / UNDERSTAND & APPLY
Input expressions belong inside the rule
To evaluate f(a + h), replace every x in the rule by the whole expression a + h. The output difference f(a + h) − f(a) equals mh for a linear function. Dividing by a nonzero input change h gives the same slope every time.
03 / UNDERSTAND & APPLY
Recognize constant rate in a table
A table is consistent with a linear model when every ratio Δoutput/Δinput is equal. Equal output differences are evidence only if the input gaps are equal. A finite table can be fit by other rules; a question must state or justify using a linear model.
ORIGINAL WORKED EXAMPLE
Unequal input gaps
A linear function has f(1) = 9 and f(5) = 21. Find f(8).
- The rate is (21 − 9)/(5 − 1) = 12/4 = 3.
- From input 5 to 8 the change is 3, so the output rises 3(3) = 9.
- Thus f(8) = 21 + 9 = 30.
- Equivalently b = 9 − 3(1) = 6, so f(x) = 3x + 6.
Answer 30
04 / UNDERSTAND & APPLY
Read slope, intercept and zero
The y-intercept is (0,f(0)); a zero is an input where f(x) = 0. For f(x) = mx + b with m ≠ 0, the zero is −b/m. Do not confuse the zero input with the output zero. The sign of m determines whether the function is increasing or decreasing.
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ORIGINAL WORKED EXAMPLE
Read the intercept meaning
A candle has height H(t) = 18 − 0.6t centimeters after t hours. Find and interpret H(5) and its zero.
- H(5) = 18 − 0.6(5) = 15, so after 5 hours its height is 15 cm.
- Solve 18 − 0.6t = 0: t = 30.
- The zero predicts that the modeled candle is used up at 30 hours. The physically meaningful domain is 0 ≤ t ≤ 30.
Answer H(5) = 15 cm; zero at t = 30 hours
05 / UNDERSTAND & APPLY
Choose the initial value carefully
Build a model using output = initial output + rate × elapsed input. The starting observation may occur at a nonzero input, so it need not be the intercept. Use f(x) = f(a) + m(x − a) when an observed point is easier than the value at zero.
ORIGINAL WORKED EXAMPLE
An observation is not an intercept
At mile marker 12, a road is 340 meters above sea level. Its elevation rises at a constant 5 meters per mile. Give E(x), with x the mile marker.
- The point is (12,340) and slope is 5 meters per mile.
- E(x) = 340 + 5(x − 12).
- Simplify: E(x) = 5x + 280.
- The coefficient 280 is the extrapolated elevation at marker 0, not the elevation at marker 12.
Answer E(x) = 5x + 280
06 / UNDERSTAND & APPLY
Changing a linear function
Adding k outside f shifts every output by k: g(x) = f(x) + k. Replacing x with x − h shifts the graph right by h: g(x) = f(x − h). With a linear rule these can look alike after simplification, so do the substitution before describing coefficients. Multiplying f(x) by c multiplies both slope and intercept by c.
ORIGINAL WORKED EXAMPLE
An input shift
Let f(x) = 2x + 3 and g(x) = f(x − 4). Find g(0) and the slope of g.
- Substitute x − 4 as one group: g(x) = 2(x − 4) + 3.
- Simplify to g(x) = 2x − 5.
- Thus g(0) = −5 and the slope remains 2.
Answer g(0) = −5; slope 2
07 / UNDERSTAND & APPLY
Compare rates and starting values separately
A larger starting value does not guarantee a larger output for all inputs. Two linear cost functions with different slopes may intersect. Solve equality to find the crossover, then use the slope or a test input to determine which model is larger on each side. If the domain is whole numbers, evaluate nearby integers.
The average rate of change between any two distinct inputs is m. An average of two output values, by contrast, is an output statistic and generally is not the slope.
08 / UNDERSTAND & APPLY
Recover an input from an output
An output request evaluates the rule; an input request solves an equation. If C(t) = 28 + 6t dollars for t hours, C(t) = 70 gives 6t = 42, so t = 7 hours. The point (7,70) lies on the cost graph and says the same thing.
A constant function f(x) = 9 has no input producing output 10 and every allowed input producing output 9. Thus “find the input” does not always have exactly one answer. A linear rule with nonzero slope has one algebraic input for a given output, but that input must still lie in its context domain.
09 / UNDERSTAND & APPLY
Use tables and graphs as checks
A calculator table can verify several input/output pairs or locate where a function changes sign. For an exact crossover or intercept, solve algebraically and substitute. Label axes with the actual variables and units; a graph that silently swaps input and output changes the meaning.
Avoid reading f(a) as an input, assuming every increase is a percent increase, or taking an output difference without dividing by its input gap.
10 / UNDERSTAND & APPLY
Original practice: understand the rule
Work with the stated linear models.
ORIGINAL PRACTICE · 1
If f(x) = −3x + 8, find f(−2).
Reveal answer & reasoning
14
Substitute −2: −3(−2) + 8 = 6 + 8 = 14.
ORIGINAL PRACTICE · 2
For f(x) = 7x − 1, find f(a + 3) − f(a).
Reveal answer & reasoning
21
An input increase of 3 changes the output by slope × change = 7 × 3.
ORIGINAL PRACTICE · 3
A linear function passes through (2,5) and (6,13). Find its rule.
Reveal answer & reasoning
f(x) = 2x + 1
Slope is 8/4 = 2; 5 = 2(2) + b gives b = 1.
ORIGINAL PRACTICE · 4
Find the zero of f(x) = 15 − 5x.
Reveal answer & reasoning
x = 3
Set the output equal to zero: 15 − 5x = 0.
ORIGINAL PRACTICE · 5
Are the points (1,3), (3,7), (6,13) consistent with one linear function?
Reveal answer & reasoning
Yes
The successive slopes are 4/2 = 2 and 6/3 = 2; all three lie on y = 2x + 1.
ORIGINAL PRACTICE · 6
If g(x) = f(x) + 6 and f(x) = −2x + 4, what is g(0)?
Reveal answer & reasoning
10
Adding 6 to the output gives g(x) = −2x + 10.
ORIGINAL PRACTICE · 7
Plans cost A(n) = 20 + 4n and B(n) = 8 + 7n dollars. At what n are they equal?
Reveal answer & reasoning
n = 4
20 + 4n = 8 + 7n gives 12 = 3n. Both cost $36 at n = 4.
ORIGINAL PRACTICE · 8
A machine makes 12 parts per minute and has already made 35 parts when timing starts. Give P(t) and interpret P(0).
Reveal answer & reasoning
P(t) = 35 + 12t; P(0) = 35 parts
The rate multiplies elapsed minutes; the existing parts are the starting output, added once.
KEEP THE BIG IDEAS
Quick revision
- Function notation names an output for an input.
- Use output change divided by input change, including unequal input gaps.
- Interpret intercepts, zeros and predictions within the model domain.
READ WITH CONTEXT
References & scope
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- College Board: SAT Math skill descriptions ↗
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