ACT Functions and Graphs: Worked Questions

Functions and graphs appear throughout the enhanced ACT Math section. The questions rarely ask for a definition; they ask you to evaluate, interpret, or transform a function in a concrete situation. This lesson explains the three skills ACT tests most often — notation, graph reading, and transformations — then walks through eight original four-choice questions. Each solution includes a second check and a diagnosis of the mistake behind every tempting wrong answer.

Function notation without fear

The notation f(x) is not a multiplication. It names the output of the function f when the input is x. To evaluate f(−2), replace every x in the rule with −2, then simplify using the ordinary order of operations. Parentheses matter, especially with negative inputs and exponents.

Reading graphs

A graph of y = f(x) is a picture of every input-output pair. To read it, locate the input value on the horizontal axis, move vertically to the graph, then read the corresponding output on the vertical axis. The same idea works in reverse: a point (a, b) on the graph tells you f(a) = b.

• Intercepts: the y-intercept is f(0), where the graph crosses the vertical axis. An x-intercept is an input value where f(x) = 0, where the graph crosses the horizontal axis.
• Slope from a graph: pick two clear points on a line, then compute (change in output) ÷ (change in input). Slope measures rate of change in context.
• Meaning in context: if the horizontal axis is time in hours and the vertical axis is dollars, a point (3, 120) means the cost is $120 after 3 hours — not $3 or 120 hours.

Transformations

Starting from a known graph y = f(x), you can build related graphs by changing the input or the output. The three transformations ACT uses most often are shifts and reflections.

Common mistakes

Most function and graph errors fall into a small set of patterns. Recognizing them in practice keeps you from repeating them on test day.

• Treating f(x) like multiplication. f(3) does not mean f times 3; it means the output when the input is 3.
• Horizontal shift direction. f(x − h) shifts right; f(x + h) shifts left. The sign inside the parentheses is reversed from the direction.
• Reading coordinates backward. In (a, b), a is the input (horizontal) and b is the output (vertical). Mixing them up turns an intercept question into a wrong answer.
• Stopping halfway through a composition. f(g(x)) means evaluate g first, then feed that result into f. Do not reverse the order or stop at g(x).
• Ignoring units and context. A slope of 2 in a distance-time graph means 2 miles per hour, not $2 or 2 hours.

Worked practice

The eight questions below are original PrepSolution practice items written in the enhanced four-choice format. Each answer was verified with two independent methods.

Question 1: Evaluate a linear function

If f(x) = 4 − 5x, what is f(3)?
A) 19
B) −11
C) 7
D) −19

Solution: replace x with 3. f(3) = 4 − 5(3) = 4 − 15 = −11.

Check another way: rewrite the function as f(x) = −5x + 4. Then f(3) = −15 + 4 = −11.

• A) 19 comes from using −3 instead of 3: 4 − 5(−3) = 19.
• C) 7 comes from adding 4 and 3 instead of multiplying 5 by 3.
• D) −19 comes from a sign error on the constant: −4 − 15 = −19.

Question 2: Function composition

If f(x) = 3x − 2 and g(x) = x², what is f(g(2))?
A) 4
B) 14
C) 16
D) 10

Solution: work from the inside out. First evaluate g(2) = 2² = 4. Then use that result as the input for f: f(4) = 3(4) − 2 = 12 − 2 = 10.

Check another way: the composition f(g(x)) means 3(x²) − 2. Substituting x = 2 gives 3(4) − 2 = 10.

• A) 4 is g(2), the inner value only; the problem asks for f(g(2)).
• B) 14 comes from computing 3(4) + 2, treating the function as f(x) = 3x + 2.
• C) 16 comes from squaring the output of f(2): f(2) = 4, then 4² = 16, which reverses the order of operations.

Question 3: Read a value from a graph

The graph of y = f(x) passes through the points shown in the table below. What is f(0)?

A) 0
B) −2
C) 3
D) 4

Solution: f(0) is the output when the input is 0. From the table, when x = 0, f(x) = 3.

Check another way: the point (0, 3) lies on the graph, which means the y-intercept is 3.

• A) 0 is an x-intercept, not the value of f(0).
• B) −2 is f(4), the output when x = 4.
• D) 4 is f(−1), the output when x = −1.

Question 4: Find an x-intercept

Using the same table of values, which of the following is an x-intercept of the graph of y = f(x)?

A) 3
B) 4
C) 0
D) −3

Solution: an x-intercept occurs where f(x) = 0. From the table, f(−3) = 0 and f(2) = 0, so −3 and 2 are x-intercepts. Of the choices given, −3 is listed.

Check another way: the graph crosses the horizontal axis at points where the y-coordinate is 0. The points (−3, 0) and (2, 0) satisfy that condition.

• A) 3 is the y-intercept f(0), not an x-intercept.
• B) 4 is the x-coordinate of a point where f(4) = −2, not 0.
• C) 0 names the y-value of every x-intercept, not the x-value the question asks for.

Question 5: Slope from two points

A straight line passes through the points (0, 2) and (6, 8). What is the slope of the line?
A) 1/6
B) 1
C) 6
D) 2

Solution: slope m = (change in y) / (change in x) = (8 − 2) / (6 − 0) = 6/6 = 1.

Check another way: using the slope-intercept form, the line is y = x + 2. When x = 6, y = 8, matching the second point.

• A) 1/6 comes from inverting the slope formula: run/rise instead of rise/run.
• C) 6 is the rise only; it ignores the run of 6 in the denominator.
• D) 2 comes from subtracting the y-intercept from the rise: (8 − 2) ÷ (6 − 0) would need a different denominator to yield 2, but this distractor catches students who guess the y-intercept.

Question 6: Identify a transformation

The graph of y = g(x) is obtained from the graph of y = f(x) by shifting 3 units to the right and reflecting across the x-axis. Which equation defines g(x)?
A) g(x) = −f(x + 3)
B) g(x) = f(x) − 3
C) g(x) = −f(x − 3)
D) g(x) = f(−x + 3)

Solution: shifting 3 units right replaces x with (x − 3), giving f(x − 3). Reflecting across the x-axis multiplies the output by −1, giving −f(x − 3).

Check another way: test a point. Suppose f(0) = 5, so (0, 5) is on y = f(x). Shifting right 3 moves it to (3, 5). Reflecting across the x-axis moves it to (3, −5). For g(3) to equal −5, the equation g(x) = −f(x − 3) gives g(3) = −f(0) = −5, which matches.

• A) −f(x + 3) shifts 3 units left, not right, before reflecting.
• B) f(x) − 3 shifts the graph down 3 units, with no reflection and no horizontal shift.
• D) f(−x + 3) reflects across the y-axis and shifts, not across the x-axis.

Question 7: Match a verbal scenario to a linear model

A plumber charges a $50 service fee plus $40 per hour. Which function gives the total cost C, in dollars, for a job that lasts h hours?
A) C(h) = 90h
B) C(h) = 50h + 40
C) C(h) = 40h + 50
D) C(h) = 50 + 40/h

Solution: the $40 per hour is the variable cost, so it is multiplied by h. The $50 service fee is a one-time fixed cost added at the end. The model is C(h) = 40h + 50.

Check another way: for a 1-hour job, the cost should be $50 + $40 = $90. Only C gives C(1) = 40(1) + 50 = 90.

• A) 90h treats the total for one hour as the hourly rate, ignoring that the $50 fee applies only once.
• B) 50h + 40 swaps the rate and the flat fee.
• D) 50 + 40/h divides the hourly rate by the number of hours, which would make longer jobs cheaper.

Question 8: Quadratic vertex from vertex form

What are the coordinates of the vertex of the graph of y = 2(x − 3)² + 7?
A) (−3, 7)
B) (3, −7)
C) (−3, −7)
D) (3, 7)

Solution: vertex form is y = a(x − h)² + k, where the vertex is (h, k). Here h = 3 and k = 7, so the vertex is (3, 7).

Check another way: the squared term is zero when x = 3, giving y = 7. Because (x − 3)² is always nonnegative and the coefficient 2 is positive, y = 7 is the minimum value, confirming the vertex.

• A) (−3, 7) comes from reading (x − 3) as a left shift instead of a right shift.
• B) (3, −7) comes from taking the opposite of the constant term k.
• C) (−3, −7) combines both sign errors.

What to study next

Use these eight questions as a focused diagnostic, not as a scaled-score prediction. A short set cannot estimate your ACT Math score; only a full official form with its own scoring key can do that.

• If notation or composition was the main miss, review the evaluating rules above, then do more problems on the free ACT practice page.
• If graph reading or slope cost you points, practice translating points, intercepts, and slope into context on the enhanced ACT math walkthrough.
• If transformations or the vertex form felt shaky, use the ACT math formulas guide to lock in the standard forms and their meanings.
• For a full schedule, see the ACT study plan.