ACT Math Shifting and Transforming Functions: 10 Practice Problems with Step-by-Step Explanations
Function transformation problems on the ACT test one core skill: reading how a function's equation changes to figure out how its graph moved. The rule that trips up the most students is that horizontal shifts work backward from what feels intuitive — replacing x with (x−h) inside a function shifts the graph right by h units, not left, even though the sign looks negative. Vertical shifts behave exactly as expected: adding a constant outside the function shifts the graph up, and subtracting shifts it down.
For most transformation problems, the fastest approach is comparing the "inside" part of the function (whatever is being done to x before any outer operations) to find the horizontal shift, and comparing the constant added or subtracted at the very end to find the vertical shift — this works whether the function is a parabola, an absolute value, a cosine curve, or a cubic. When a leading coefficient's sign flips between the original and transformed function (like from positive to negative), that indicates a reflection over the x-axis in addition to any shifts. Translating individual points or lines follows the same logic, just applied directly to specific coordinates rather than to a general equation.
Common traps include getting the direction of a horizontal shift backward (treating x+h as a right shift instead of left, or vice versa), missing a required reflection when only checking the shift and not the sign of the leading coefficient, and on problems that track a specific point through a transformed function, applying the transformations in the wrong order or forgetting one of them entirely. Work through the 10 problems below, then click to reveal each step-by-step explanation.
On this page:
- Problem 1 — Translating a point
- Problem 2 — Vertical shift from a constant
- Problem 3 — Vertical shift of a trig function
- Problem 4 — Combined horizontal and vertical shift
- Problem 5 — Shift from a vertex comparison
- Problem 6 — Building a shifted function
- Problem 7 — Shift with no reflection
- Problem 8 — Shift with a reflection
- Problem 9 — Tracking a point through a transformation
- Problem 10 — Translating a line
Practice Problems
In the standard (x,y) coordinate plane, point A has coordinates (5,−9). Point A is translated 4 units to the right and 5 units up to Point B. What are the coordinates of Point B?
- A) (9,−14)
- B) (9,−4)
- C) (5,−4)
- D) (0,−5)
Add the horizontal shift to the x-coordinate: 5+4=9. Add the vertical shift to the y-coordinate: −9+5=−4.
Point B is (9,−4).
In the standard (x,y) coordinate plane, the graph of the function y=(x+11)2−20 is shifted such that its image is at y=(x+11)2−9. Which of the following describes this shift?
- A) Down 9 units.
- B) Right 11 units.
- C) Left 11 units.
- D) Up 11 units.
The "inside" part of the function, (x+11), is unchanged, so there's no horizontal shift.
The constant changes from −20 to −9, an increase of 11, meaning the graph shifted up 11 units.
In the standard (x,y) coordinate plane, the graph of the function y=2cos(x)+3 undergoes a single translation such that the equation of its image is y=2cos(x)−7. Which of the following describes this translation?
- A) Left 10 coordinate units
- B) Down 4 coordinate units
- C) Up 10 coordinate units
- D) Down 10 coordinate units
The "inside" of the cosine function is unchanged (both have just x), so there's no horizontal shift.
The constant changes from +3 to −7, a decrease of 10, meaning the graph shifted down 10 units.
h(x) = x2−5
g(x) = (x−2)2−1
Which of the following correctly describes the shift required to transform h(x) into g(x)?
- A) Shift h(x) right 4 units and down 2 units.
- B) Shift h(x) right 2 units and down 4 units.
- C) Shift h(x) right 2 units and up 4 units.
- D) Shift h(x) left 2 units and down 4 units.
The inside changes from x to (x−2), meaning a shift right by 2 units.
The constant changes from −5 to −1, an increase of 4, meaning a shift up by 4 units.
In the standard (x,y) coordinate plane, the graph of the function y=|x+2|+6 undergoes two translations such that the equation of its image is y=|x+4|+9. Which of the following describes these translations?
- A) Left 2 units and down 3 units
- B) Right 2 units and down 3 units
- C) Left 2 units and up 3 units
- D) Right 2 units and up 3 units
Find each function's vertex. For y=|x+2|+6, the vertex is at (−2,6). For y=|x+4|+9, the vertex is at (−4,9).
Compare the vertices: x changes from −2 to −4 (a shift left 2 units), and y changes from 6 to 9 (a shift up 3 units).
f(x) = (x+1)3+2
Which of the following functions g(x) shifts f(x) up by 4 units and left by 1 unit?
- A) g(x) = x3+6
- B) g(x) = (x+5)3+3
- C) g(x) = (x+2)3−2
- D) g(x) = (x+2)3+6
To shift left by 1 unit, replace x with (x+1) inside f: f(x+1) = ((x+1)+1)3+2 = (x+2)3+2.
To shift up by 4 units, add 4 to the entire result: (x+2)3+2+4 = (x+2)3+6.
In the standard (x,y) coordinate plane, given Parabola P with equation y=−2x2, Parabola Q is the image of Parabola P after a shift of 5 coordinate units up and 8 coordinate units left. Parabola Q has which of the following equations?
- A) y=−2(x+5)2−8
- B) y=−2(x+8)2+5
- C) y=−2(x−5)2−8
- D) y=−2(x+8)2−5
To shift left by 8 units, replace x with (x+8) inside the function: y=−2(x+8)2.
To shift up by 5 units, add 5 to the entire result: y=−2(x+8)2+5.
g(x) = −(x−2)2+9
h(x) = −(x+2)2+11
Which of the following transformations are required to turn h(x) into g(x)?
- A) Shift 2 units left and reflect over the x-axis.
- B) Shift 4 units left and 2 units up.
- C) Shift 4 units right and reflect over the x-axis.
- D) Shift 4 units right and 2 units down.
Both functions already have a negative leading coefficient (−1), so no reflection is needed. The inside changes from (x+2) to (x−2), a shift right by 4 units.
The constant changes from 11 to 9, a decrease of 2, meaning a shift down 2 units.
The function f(x), when graphed in the standard (x,y) coordinate plane, includes point Z at (0,5). When function is −f(x−3)+4 is drawn, point Z is located at:
- A) (−3,9)
- B) (0,−9)
- C) (3,−1)
- D) (−1,4)
Since Z=(0,5) means f(0)=5, find the new x-coordinate by setting the inside expression equal to 0: x−3=0, so x=3.
Find the new y-coordinate by applying the reflection and vertical shift to the original output: −f(0)+4 = −5+4 = −1. The new location of Z is (3,−1).
Line k contains the points (14,166) and (19,206). Line a is the result of translating line k down 7 units in the xy-plane. What is the x-coordinate of the x-intercept of line a?
- A) −61/8
- B) −27/4
- C) −47/8
- D) −31/6
Find line k's slope: (206−166)/(19−14) = 40/5 = 8. Using point-slope form with (14,166): y−166=8(x−14), which simplifies to y=8x+54.
Shifting down 7 units gives line a: y=8x+47. Find the x-intercept by setting y=0: 0=8x+47, so x=−47/8.
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