Transformations of Functions

Unit 1 · Polynomial and Rational Functions

What AP Precalc asks here

Transformations rewrite a graph in terms of the underlying function $f$ . Outside-the-function changes are vertical: $f(x) + k$ shifts up by $k$ , $a \cdot f(x)$ stretches vertically by factor $a$ , and $-f(x)$ reflects across the x-axis. Inside-the-function changes are horizontal but counterintuitive: $f(x - h)$ shifts the graph right by $h$ , $f(bx)$ compresses horizontally by factor $b$ , and $f(-x)$ reflects across the y-axis.

Vertical transformations (outside)

Shift

g(x) = f(x) + k \;\Rightarrow\; \text{shift up by } k

Stretch

g(x) = a \cdot f(x) \;\Rightarrow\; \text{vertical stretch by } a

Reflect over x-axis

g(x) = -f(x)

Horizontal transformations (inside, counterintuitive)

Shift

g(x) = f(x - h) \;\Rightarrow\; \text{shift right by } h

Compress

g(x) = f(bx) \;\Rightarrow\; \text{horizontal compression by } b

Reflect over y-axis

g(x) = f(-x)

Caution: Inside-the-function shifts are opposite the sign convention.

f(x - 3)

shifts the graph right by 3, not left.

Type 1

Single shift, stretch, or reflection

Apply one transformation. Pull values directly from the given f data and apply the rule.

Example 1

g(x) = f(x - 3)

and the graph of

f

has a y-intercept at

(0, 2)

, where is the corresponding point on

g

? (A)

(-3, 2)

(B)

(0, 5)

(C)

(3, 2)

(D)

(0, -1)

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Type 2

Combined transformation

Apply transformations in order: horizontal shift first, then horizontal stretch/reflect, then vertical stretch/reflect, then vertical shift.

Example 2

g(x) = 3 f(x - 2) + 1

and

f

has its maximum value of

4

x = 5

, where is the maximum of

g

? (A)

(3, 12)

(B)

(7, 12)

(C)

(7, 13)

(D)

(5, 13)

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Type 3

Inside-vs-outside rules

Distinguish vertical and horizontal effects. Watch for sign and reciprocal patterns inside the function.

Example 3

g(x) = f(3x)

and

f

has zeros at

x = 6

and

x = -3

, find the zeros of

g

. (A)

x = 2

and

x = -1

(B)

x = 18

and

x = -9

(C)

x = 6

and

x = -3

(D)

x = -2

and

x = 1

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