Precalculus Transformations of Graphs Lesson (Free PDF)
This precalculus transformations lesson for Precalculus covers transformations of graphs (Common Core F-BF.B.3, F-IF.C.7b), with a free PDF to print.
Transformations move or reshape a parent graph. In y = a f(b(x – h)) + k, h shifts right, k shifts up, a stretches vertically and reflects over the x-axis when negative, and b compresses horizontally. Inside changes act opposite to their sign, so f(x – 3) shifts the graph right 3 units.
Precalculus · Common Core State Standards: F-BF.B.3, F-IF.C.7b
Key facts in this precalculus transformations lesson
- f(x) + k shifts up k; f(x – h) shifts right h.
- -f(x) reflects over the x-axis; f(-x) reflects over the y-axis.
- a f(x) stretches vertically by |a|; f(bx) compresses horizontally by 1/|b|.
- A point (p, q) on f becomes (p + h, a q + k) on a f(x – h) + k.
- Even functions satisfy f(-x) = f(x); odd functions satisfy f(-x) = -f(x).
What are the basic transformations of a graph?
Start from a parent function such as \(f(x)=x^{2}\), \(|x|\), \(\sqrt{x}\), \(x^{3}\) or \(\frac{1}{x}\). The graph of \(y=a\,f(b(x-h))+k\) is the parent graph moved and reshaped. Changes outside \(f\) act on \(y\) the way you expect. Changes inside the parentheses act on \(x\) in the opposite way from what the sign suggests.
| Equation | Effect on the graph |
|---|---|
| \(y=f(x)+k\) | Shift up \(k\) units (down if \(k< 0\)) |
| \(y=f(x-h)\) | Shift right \(h\) units (left if \(h< 0\)) |
| \(y=-f(x)\) | Reflect over the \(x\)-axis |
| \(y=f(-x)\) | Reflect over the \(y\)-axis |
| \(y=a\,f(x)\) | Vertical stretch if \(|a|>1\), compression if \(0<|a|< 1\) |
| \(y=f(bx)\) | Horizontal compression by \(\frac{1}{b}\) if \(|b|>1\), stretch if \(0<|b|< 1\) |
How do you move a point through several transformations?
Apply the changes to each coordinate separately. A point \((p,q)\) on \(f\) moves to \((p+h,\ a q+k)\) on \(y=a f(x-h)+k\). Use the order below when more than one change is present.
- Horizontal changes first: stretch or compress, reflect over the \(y\)-axis, then shift left or right.
- Vertical changes next: stretch or compress, reflect over the \(x\)-axis, then shift up or down.
How do reflections and stretches change a graph?
The sign of \(a\) controls the reflection, and \(|a|\) controls the stretch. A negative inside the function flips left to right, while a negative outside flips top to bottom.
What are even and odd functions?
A function is even if \(f(-x)=f(x)\) for all \(x\) (symmetric about the \(y\)-axis), and odd if \(f(-x)=-f(x)\) (symmetric about the origin). For example \(x^{2}\) is even, \(x^{3}\) is odd, and \(x^{2}+x\) is neither.
Precalculus transformations lesson: questions and answers
Why does f(x – 3) shift right instead of left?
The output f(x – 3) at x = 5 equals the old output at x = 2, so every feature moves 3 units to the right. The input must be 3 larger to reach the same value.
What is the difference between f(-x) and -f(x)?
f(-x) reflects the graph over the y-axis. -f(x) reflects it over the x-axis. They are equal only for special functions.
In what order do you apply transformations?
Do horizontal changes first (stretch, reflect, shift) and vertical changes next. For y = -2 sqrt(x – 3) + 1, shift right 3, stretch by 2, reflect over the x-axis, then shift up 1.
How can you tell if a function is even or odd?
Compute f(-x). If it equals f(x) the function is even. If it equals -f(x) the function is odd. If neither holds, the function is neither.
How do transformations change the domain and range?
Horizontal moves shift the domain and vertical moves shift the range. Reflections and stretches can flip or scale them. The square root shifted left 4 has domain x >= -4.
How to use this precalculus transformations lesson
Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.
Download the precalculus transformations lesson as a PDF with the button below and keep it with your notes.
Download the precalculus transformations lesson as a PDF with the button below and keep it with your notes.
All Precalculus chapters: Precalculus math.
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