2.4 Transformations of Functions

2.4.1 Learning Objectives

By the end of this section, you should be able to:

describe vertical and horizontal shifts;
reflect graphs across the coordinate axes;
apply vertical and horizontal stretches and compressions;
combine transformations in an appropriate order;
write transformed function rules; and
identify even and odd functions from formulas and graphs.

2.4.2 Vertical Shifts

Adding a constant outside a function shifts its graph vertically:

\[ g(x)=f(x)+k. \]

If \(k>0\), the graph moves upward \(k\) units.
If \(k<0\), the graph moves downward \(|k|\) units.

Every point \((x,y)\) on \(f\) becomes \((x,y+k)\) on \(g\).

Example 2.16 Applying a Vertical Shift

Describe the transformation from \(f(x)=x^2\) to

\[ g(x)=x^2-5. \]

Solution

Subtracting \(5\) outside the function shifts the parabola \(\boxed{5\text{ units downward}}\). Its vertex moves from \((0,0)\) to \((0,-5)\).

Exercise 2.15 Describe the transformation from \(f(x)=\sqrt{x}\) to \(g(x)=\sqrt{x}+3\).

2.4.3 Horizontal Shifts

Changing the input shifts a graph horizontally:

\[ g(x)=f(x-h). \]

\(f(x-h)\) shifts the graph right \(h\) units.
\(f(x+h)\) shifts the graph left \(h\) units.

The direction appears opposite to the sign because the transformation changes which input produces each original output.

Example 2.17 Applying a Horizontal Shift

Describe the transformation from \(f(x)=|x|\) to

\[ g(x)=|x+4|. \]

Solution

The \(+4\) occurs inside the function, so the graph shifts \(\boxed{4\text{ units left}}\). The vertex moves from \((0,0)\) to \((-4,0)\).

Exercise 2.16 Describe the transformation from \(f(x)=x^3\) to \(g(x)=(x-2)^3\).

2.4.4 Reflections

Multiplying the output by \(-1\) reflects a graph across the \(x\)-axis:

\[ g(x)=-f(x). \]

Replacing \(x\) by \(-x\) reflects a graph across the \(y\)-axis:

\[ h(x)=f(-x). \]

Example 2.18 Reflecting a Square-Root Graph

Describe the transformations from \(f(x)=\sqrt{x}\) to

\[ g(x)=-\sqrt{-x}. \]

Solution

The replacement \(x\mapsto -x\) reflects the graph across the \(y\)-axis. The negative sign outside then reflects it across the \(x\)-axis. The graph begins at the origin and extends left and downward.

Exercise 2.17 Describe the transformation from \(f(x)=x^2\) to \(g(x)=-x^2\).

2.4.5 Vertical Stretches and Compressions

For

\[ g(x)=af(x), \]

the factor \(a\) multiplies every output.

If \(|a|>1\), the graph is stretched vertically by factor \(|a|\).
If \(0<|a|<1\), the graph is compressed vertically by factor \(|a|\).
If \(a<0\), a reflection across the \(x\)-axis also occurs.

Example 2.19 Comparing Vertical Scale Factors

Describe \(g(x)=3|x|\) and \(h(x)=\frac12|x|\) relative to \(f(x)=|x|\).

Solution

The graph of \(g\) is stretched vertically by a factor of \(\boxed{3}\). The graph of \(h\) is compressed vertically by a factor of \(\boxed{1/2}\).

Exercise 2.18 Describe \(g(x)=-2f(x)\) relative to \(f\).

2.4.6 Horizontal Stretches and Compressions

For

\[ g(x)=f(bx), \]

the horizontal scale factor is \(1/|b|\).

If \(|b|>1\), the graph is compressed horizontally by factor \(1/|b|\).
If \(0<|b|<1\), the graph is stretched horizontally by factor \(1/|b|\).
If \(b<0\), a reflection across the \(y\)-axis also occurs.

Example 2.20 Applying a Horizontal Scale Factor

Describe \(g(x)=f(4x)\) relative to \(f(x)\).

Solution

Inputs one quarter as large produce the original outputs, so the graph is compressed horizontally by a factor of \(\boxed{1/4}\).

Exercise 2.19 Describe \(h(x)=f(x/3)\) relative to \(f(x)\).

2.4.7 Combining Transformations

A transformed function often has the form

\[ g(x)=af(b(x-h))+k. \]

The parameters describe:

a horizontal shift by \(h\);
a horizontal scale factor of \(1/|b|\), with a \(y\)-axis reflection if \(b<0\);
a vertical scale factor of \(|a|\), with an \(x\)-axis reflection if \(a<0\); and
a vertical shift by \(k\).

For point mapping, a point \((x,y)\) on \(f\) corresponds to

\[ \left(\frac{x}{b}+h,ay+k\right) \]

on \(g\).

Example 2.21 Describing Several Transformations

Describe the graph of

\[ g(x)=-2(x-3)^2+5 \]

relative to \(f(x)=x^2\).

Solution

Starting with \(y=x^2\):

shift right \(3\) units;
stretch vertically by factor \(2\);
reflect across the \(x\)-axis; and
shift upward \(5\) units.

The vertex is \(\boxed{(3,5)}\), and the parabola opens downward.

Exercise 2.20 Describe \(g(x)=\frac12|x+1|-4\) relative to \(f(x)=|x|\).

2.4.8 Even and Odd Functions

A function is even if

\[ f(-x)=f(x) \]

for every \(x\) in its domain. Its graph is symmetric about the \(y\)-axis.

A function is odd if

\[ f(-x)=-f(x). \]

Its graph is symmetric about the origin. A function may be neither even nor odd.

Example 2.22 Testing Algebraically for Symmetry

Classify

\[ f(x)=x^4-3x^2+2 \]

as even, odd, or neither.

Solution

\[ \begin{aligned} f(-x) &=(-x)^4-3(-x)^2+2\\ &=x^4-3x^2+2\\ &=f(x). \end{aligned} \]

Therefore, \(f\) is \(\boxed{\text{even}}\).

Exercise 2.21 Classify \(g(x)=x^3-2x\) as even, odd, or neither.

2.4.9 Desmos Graphing Activity

In Desmos, enter a base function as \(f(x)\), then graph transformed versions using sliders for \(a\), \(b\), \(h\), and \(k\):

\[ g(x)=a f(b(x-h))+k. \]

Change one parameter at a time. Record how it affects the position, orientation, and scale of the graph. Then test combinations of parameters and predict the graph before moving the sliders.

2.4.10 Conceptual Takeaways

Changes outside a function affect outputs and therefore act vertically.
Changes inside a function affect inputs and therefore act horizontally.
Horizontal shifts and scale factors appear opposite to their algebraic signs or sizes.
Negative factors create reflections.
Several transformations can be combined in one function rule.
Even functions have \(y\)-axis symmetry, while odd functions have origin symmetry.

2.4.11 Skills You Should Be Able to Do

Describe vertical and horizontal shifts.
Apply reflections across the coordinate axes.
Distinguish vertical stretches from compressions.
Distinguish horizontal stretches from compressions.
Write and interpret functions of the form \(af(b(x-h))+k\).
Map important points through a transformation.
Test functions algebraically for even or odd symmetry.
Explore transformations using Desmos.

2.4.12 Practice Problems with Solutions

Describe the transformation from \(f(x)=x^2\) to \(g(x)=x^2+7\).

Show Solution

Adding \(7\) outside the function shifts the graph \(\boxed{7\text{ units upward}}\).
Describe the transformation from \(f(x)=\sqrt{x}\) to \(g(x)=\sqrt{x-5}\).

Show Solution

Replacing \(x\) with \(x-5\) shifts the graph \(\boxed{5\text{ units right}}\).
Describe \(g(x)=-f(x+2)\) relative to \(f(x)\).

Show Solution

The graph shifts \(2\) units left and reflects across the \(x\)-axis.
Describe \(h(x)=4f(x)\) relative to \(f(x)\).

Show Solution

The graph is stretched vertically by a factor of \(\boxed{4}\).
Describe \(p(x)=f(2x)\) relative to \(f(x)\).

Show Solution

The graph is compressed horizontally by a factor of \(\boxed{1/2}\).
State the vertex and describe the transformations of

\[ g(x)=3(x+2)^2-4. \]

Show Solution

Relative to \(y=x^2\), shift left \(2\), stretch vertically by factor \(3\), and shift down \(4\). The vertex is

\[ \boxed{(-2,-4)}. \]
Write a function obtained by reflecting \(f(x)=|x|\) across the \(x\)-axis, shifting it right \(3\), and shifting it up \(2\).

Show Solution

The reflection gives \(-|x|\), the right shift replaces \(x\) by \(x-3\), and the upward shift adds \(2\):

\[ \boxed{g(x)=-|x-3|+2}. \]
Classify \(f(x)=x^5+3x\) as even, odd, or neither.

Show Solution

\[ f(-x)=(-x)^5+3(-x)=-x^5-3x=-f(x). \]

Therefore, \(f\) is \(\boxed{\text{odd}}\).
Classify \(g(x)=x^2+x\) as even, odd, or neither.

Show Solution

\[ g(-x)=x^2-x. \]

This is neither \(g(x)=x^2+x\) nor \(-g(x)=-x^2-x\). Therefore, \(g\) is \(\boxed{\text{neither even nor odd}}\).