Transformations of functions — AP Precalculus Study Guide

For: AP Precalculus candidates sitting AP Precalculus.

Covers: vertical and horizontal translations, reflections over coordinate axes, vertical and horizontal stretches and compressions, combined transformations, order of transformation operations, and transformations of polynomial and rational functions.

You should already know: function notation, evaluating functions for given inputs, graphing basic polynomial and rational functions.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the AP Precalculus style for educational use. They are not reproductions of past College Board / Cambridge / IB papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official mark schemes for grading conventions.

1. What Is Transformations of functions?

Function transformations are a set of rules that describe how to modify the graph or equation of a known "parent function" $f(x)$ to produce a new related function. This skill lets you analyze complex functions by connecting them to simpler, familiar parents, rather than building graphs from scratch. According to the AP Precalculus Course and Exam Description (CED), Unit 1: Polynomial and Rational Functions makes up 27–30% of the total exam score, and transformations are a core skill assessed in both multiple-choice (MCQ) and free-response (FRQ) sections. Common exam questions ask you to match transformed equations to graphs, identify transformations from a given equation, or apply transformations to contextual functions. Synonyms for this topic include graph transformations and function shifts. It is one of the most frequently tested foundational skills in the first unit of the course.

2. Rigid Transformations: Translations and Reflections

Rigid transformations are transformations that only change the position of the parent function's graph, not its shape or size. The general form for a rigid transformation is: $$g(x)=f(x-h)+k$$ The parameter $k$ controls vertical translation: adding $k$ to the output of $f(x)$ shifts every point $(x,y)$ on the original graph to $(x,y+k)$. This means if $k>0$, the graph shifts up $k$ units, and if $k<0$, it shifts down $|k|$ units. The parameter $h$ controls horizontal translation: replacing the input $x$ with $x-h$ shifts every point $(x,y)$ on the original graph to $(x+h,y)$. A common point of confusion is the sign: if $h>0$, the graph shifts right $h$ units, because to get the same output value $f(0)$, you need to input $x=h$ into $f(x-h)$, so the point that was at $x=0$ is now at $x=h$. Rigid transformations also include reflections: $g(x)=-f(x)$ reflects the graph over the $x$-axis (flips all $y$-values), while $g(x)=f(-x)$ reflects over the $y$-axis (flips all $x$-values).

Worked Example

Problem: The parent function $f(x)=x^2$ has a vertex at $(0,0)$. Give the equation of the function after a translation 4 units left, 1 unit up, then a reflection over the $x$-axis. What are the coordinates of the new vertex?

1. First, apply the translation: 4 units left means $h=-4$, so $x-h=x-(-4)=x+4$. 1 unit up means $k=1$, so after translation we get $f(x+4)+1=(x+4{)}^2+1$.
2. Apply reflection over the $x$-axis: multiply the entire function by $-1$ to flip the sign of all outputs, giving $g(x)=-\left((x+4{)}^2+1\right)=-(x+4{)}^2-1$.
3. The original vertex at $(0,0)$ is shifted 4 left and 1 up to $(-4,1)$.
4. Reflection over the $x$-axis flips the sign of the $y$-coordinate, so the new vertex is $(-4,-1)$.

Exam tip: Always rewrite horizontal translations in the standard $f(x-h)$ form to confirm the shift direction. For example, rewrite $f(x+5)$ as $f(x-(-5))$ to avoid misreading it as a right shift.

3. Non-Rigid Transformations: Stretches and Compressions

Non-rigid transformations change the shape and size of the parent function's graph, rather than just its position. The general form for scaling transformations is $g(x)=af(bx)$, where $a$ controls vertical scaling and $b$ controls horizontal scaling. For vertical scaling: multiplying the output of $f(x)$ by $a$ scales every $y$-value by $a$. If $|a|>1$, this is a vertical stretch by a factor of $|a|$ (the graph gets taller). If $0<|a|<1$, this is a vertical compression by a factor of $|a|$ (the graph gets shorter). If $a$ is negative, the scaling also includes a reflection over the $x$-axis. For horizontal scaling: replacing the input $x$ with $bx$ scales every $x$-value by $\frac{1}{|b|}$. If $|b|>1$, this is a horizontal compression by a factor of $\frac{1}{|b|}$ (the graph gets narrower horizontally). If $0<|b|<1$, this is a horizontal stretch by a factor of $\frac{1}{|b|}$ (the graph gets wider horizontally). If $b$ is negative, the scaling also includes a reflection over the $y$-axis. The reciprocal rule for horizontal scaling is the most commonly tested rule for this subtopic.

Worked Example

Problem: Given parent function $f(x)=x^3$, write the equation of the transformed function after a horizontal compression by a factor of $\frac{1}{2}$ and a vertical stretch by a factor of 4. What is the value of the transformed function at $x=3$?

1. A horizontal compression by factor $\frac{1}{2}$ means the scale factor $\frac{1}{|b|}=\frac{1}{2}$, so solving for $b$ gives $|b|=2$. There is no reflection, so $b=2$.
2. A vertical stretch by factor 4 means $a=4$, with no reflection, so the general form is $g(x)=4f(2x)$.
3. Substitute $f(z)=z^3$ to get $g(x)=4(2x{)}^3=4(8x^3)=32x^3$.
4. Evaluate at $x=3$: $g(3)=32(3{)}^3=32(27)=864$.

Exam tip: Remember the reciprocal rule for horizontal scaling: the scale factor is always the reciprocal of the coefficient of $x$ inside the function. Never directly use the coefficient as the scale factor for horizontal transformations.

4. Combined Transformations and Order of Operations

When multiple transformations are applied to a parent function, they must be applied in the correct order to get the right equation and graph. The standard form of any fully transformed function is: $$g(x)=a\cdot f\left(b(x-h)\right)+k$$ Order of operations follows the same PEMDAS rules you use to evaluate $g(x)$ for a given input: first process operations on the input $x$ (inside the function), then process operations on the output of $f(x)$ (outside the function). The correct sequence is:

1. Horizontal transformations: shift by $h$, then scale/reflect by $b$ (because you subtract $h$ before multiplying by $b$ inside the parentheses)
2. Vertical transformations: scale/reflect by $a$, then shift by $k$ (because you multiply the output by $a$ before adding $k$) The most common mistake here is failing to factor out $b$ from the input term before identifying $h$, which leads to incorrect horizontal shift values.

Worked Example

Problem: Rewrite the rational function $g(x)=\frac{2x+1}{x+3}$ as a transformation of the parent function $f(x)=\frac{1}{x}$, then list all transformations in order.

1. Rewrite the numerator to separate the constant term: $2x+1=2(x+3)-5$.
2. Simplify the function: $g(x)=\frac{2(x+3)-5}{x+3}=2-\frac{5}{x+3}=-5\cdot \frac{1}{x+3}+2$.
3. Write in standard form: $g(x)=-5\cdot f\left(1(x-(-3))\right)+2$, so $a=-5$, $b=1$, $h=-3$, $k=2$.
4. List transformations in order: (1) Shift $f(x)=\frac{1}{x}$ 3 units left (no horizontal scaling/reflect), (2) Stretch vertically by a factor of 5, reflect over the $x$-axis, then shift up 2 units. Verify: the point $(1,1)$ on $f(x)$ becomes $(-2,-3)$ after transformations, and $g(-2)=\frac{2(-2)+1}{-2+3}=\frac{-3}{1}=-3$, which matches.

Exam tip: Always factor the coefficient of $x$ out of the input term before identifying the horizontal shift. For example, $f(3x+9)=f(3(x+3))$, which is a 3-unit left shift, not a 9-unit left shift.

5. Common Pitfalls (and how to avoid them)

• Wrong move: Interpreting $f(x+4)$ as a 4-unit shift right instead of 4 units left. Why: Students associate positive numbers with right movement, and forget the standard form uses $f(x-h)$, so a positive shift inside the function is actually negative $h$. Correct move: Always rewrite the input as $x-h$ to confirm: $f(x+4)=f(x-(-4))$, so $h=-4$, which is a 4-unit left shift.
• Wrong move: Interpreting $f(4x)$ as a horizontal stretch by a factor of 4 instead of a horizontal compression by a factor of 4. Why: Students directly match the coefficient $b$ to the scale factor, instead of using the reciprocal rule for horizontal transformations. Correct move: For input coefficient $b$, the horizontal scale factor is always $\frac{1}{|b|}$, so $b=4$ gives a scale factor of $\frac{1}{4}$, which is a compression.
• Wrong move: Calculating a 6-unit left shift for $f(2x+6)$, instead of a 3-unit left shift. Why: Students do not factor out the coefficient of $x$ before reading the shift value $h$. Correct move: Always factor the coefficient of $x$ inside the function: $f(2x+6)=f(2(x+3))=f(2(x-(-3)))$, so $h=-3$, which is a 3-unit left shift.
• Wrong move: Applying vertical shift before vertical stretch for $2f(x)+3$, leading to incorrect output values. Why: Students forget order of operations, and do addition before multiplication. Correct move: Always apply stretches/compressions/reflections (multiplication steps) before vertical shifts (addition steps), following PEMDAS order.
• Wrong move: Confusing $-f(x)$ as a reflection over the $y$-axis instead of the $x$-axis. Why: Students mix up whether the negative sign applies to the input (inside $f$) or output (outside $f$). Correct move: Negative outside $f$ flips $y$-values → reflection over $x$-axis; negative inside $f$ flips $x$-values → reflection over $y$-axis.

6. Practice Questions (AP Precalculus Style)

Question 1 (Multiple Choice)

The graph of $f(x)=x^4-2x^2$ is shifted 1 unit right, reflected over the $y$-axis, then shifted 3 units down. Which of the following is the equation of the transformed function? A) $g(x)=(-x-1{)}^4-2(-x-1{)}^2+3$ B) $g(x)=-(x-1{)}^4+2(x-1{)}^2-3$ C) $g(x)=(-(x-1){)}^4-2(-(x-1){)}^2-3$ D) $g(x)=(-x+1{)}^4-2(-x+1{)}^2+3$

Worked Solution: We apply transformations in the order given in the problem. First, shifting 1 unit right replaces $x$ with $x-1$ in $f(x)$, giving $f(x-1)$. Next, reflecting over the $y$-axis replaces all inputs $x$ with $-x$, so we substitute $x\to -x$ to get $f(-(x-1))$. Finally, shifting 3 units down subtracts 3 from the entire function, giving $g(x)=f(-(x-1))-3$. Substituting $f(z)=z^4-2z^2$ gives $g(x)=(-(x-1){)}^4-2(-(x-1){)}^2-3$, which matches option C. The correct answer is C.

Question 2 (Free Response)

Let the parent function be $f(x)=x^2$, and let $g(x)=2x^2-12x+7$. (a) Rewrite $g(x)$ in the form $g(x)=af(x-h)+k$, and identify all transformations applied to $f(x)$ to get $g(x)$. (b) The vertex of $f(x)=x^2$ is at $(0,0)$. What are the coordinates of the vertex of $g(x)$? (c) Verify your answer by calculating the vertex coordinates using the quadratic vertex formula $x=-\frac{b}{2a}$ for $g(x)=a{x}^2+bx+c$.

Worked Solution: (a) Complete the square to rewrite $g(x)$: $$ \begin{align*} g(x) &= 2x^2 - 12x + 7 \ &= 2(x^2 - 6x) + 7 \ &= 2\left((x^2 - 6x + 9) - 9\right) + 7 \ &= 2(x - 3)^2 - 18 + 7 = 2(x - 3)^2 - 11 \end{align*} $$ So $g(x)=2f(x-3)-11$. Transformations: shift $f(x)$ 3 units right, stretch vertically by a factor of 2, then shift 11 units down. (b) Original vertex at $(0,0)$. Shifting 3 right and 11 down gives $(3,-11)$; vertical stretch does not change the $x$-coordinate of the vertex, so the final vertex is $(3,-11)$. (c) For $g(x)=2x^2-12x+7$, $a=2$, $b=-12$, so $x=-\frac{-12}{2(2)}=3$. Substitute back to find $y=2(3{)}^2-12(3)+7=18-36+7=-11$, which confirms the vertex is $(3,-11)$.

Question 3 (Application / Real-World Style)

A local bakery models its monthly profit $P(t)$, in thousands of dollars, $t$ months after January 2023, as $P(t)=-0.4t(t-10)$, a quadratic function peaking at 5 months. In 2024, the bakery shifts its peak profit to 2 months earlier than in 2023, and its maximum profit decreases by 2 thousand dollars compared to 2023. Write the function $Q(s)$ that models monthly profit $s$ months after January 2024, and find the maximum profit for 2024.

Worked Solution: A peak 2 months earlier means a horizontal shift 2 units left, so $h=-2$, and we replace $t$ with $s+2$. A maximum profit decrease of 2 thousand dollars means a vertical shift down 2 units, so subtract 2 from the output. The transformed function is: $$ \begin{align*} Q(s) &= -0.4(s + 2)(s + 2 - 10) - 2 \ &= -0.4(s + 2)(s - 8) - 2 \end{align*} $$ The original 2023 maximum profit is $P(5)=-0.4(5)(-5)=10$ thousand dollars. After a 2 thousand dollar decrease, the new maximum profit is $10-2=8$ thousand dollars. In context, the bakery's maximum monthly profit in 2024 is $\$8,000$, occurring 3 months after January 2024.

7. Quick Reference Cheatsheet

8. What's Next

Immediately after mastering transformations of functions, you will apply this skill to graphing and analyzing quadratic, cubic, and general polynomial functions in the remainder of Unit 1. Transformations let you quickly sketch the graph of any polynomial written in vertex or factored form by relating it to a simple parent function, which saves critical time on both MCQ and FRQ sections of the exam. This topic also feeds into later units: when you study rational functions, you will use transformations to shift and scale parent reciprocal functions to graph complex rational expressions. Transformations are also a core foundation for studying trigonometric functions in Unit 3, where period and phase shift are just horizontal stretch and translation, respectively. Without mastering the order and sign rules for transformations, you will struggle to correctly identify key features like vertices, asymptotes, and extrema of more complex functions.

Graphing polynomial functions Rational function graphs and asymptotes Transformations of trigonometric functions