Graphing Techniques: Stretching, Shrinking, Reflecting, Symmetry, and Translations

Chapter 2: Graphs and Functions

Section 2.7: Graphing Techniques

This section explores the fundamental techniques for transforming the graphs of functions, including stretching, shrinking, reflecting, symmetry, and translations. Mastery of these concepts is essential for understanding how algebraic changes affect the graphical representation of functions.

Stretching and Shrinking

Vertical Stretching and Shrinking

Vertical transformations change the y-values of a function. If $f(x)$ is a function and $a > 0$, then $g(x) = a f(x)$ stretches or shrinks the graph vertically:

• Vertical Stretch: If $a > 1$, the graph is stretched away from the x-axis.

• Vertical Shrink: If $0 < a < 1$, the graph is compressed toward the x-axis.

For every point $(x, y)$ on the graph of $f(x)$, the corresponding point on $g(x)$ is $(x, a y)$.

x	$f(x) = |x|$	$g(x) = 2|x|$
-2	2	4
-1	1	2
0	0	0
1	1	2
2	2	4

x	$f(x) = |x|$	$h(x) = \frac{1}{2}|x|$
-2	2	1
-1	1	\frac{1}{2}
0	0	0
1	1	\frac{1}{2}
2	2	1

Horizontal Stretching and Shrinking

Horizontal transformations affect the x-values. For $g(x) = f(a x)$:

• Horizontal Stretch: If $0 < a < 1$, the graph is stretched away from the y-axis.

• Horizontal Shrink: If $a > 1$, the graph is compressed toward the y-axis.

For every point $(x, y)$ on the graph of $f(x)$, the corresponding point on $g(x)$ is $(\frac{x}{a}, y)$.

Reflecting

Reflection Across the x-axis and y-axis

Reflection creates a mirror image of a graph across a specified axis:

• Across the x-axis: $g(x) = -f(x)$ reflects $f(x)$ over the x-axis. Each point $(x, y)$ becomes $(x, -y)$.

• Across the y-axis: $g(x) = f(-x)$ reflects $f(x)$ over the y-axis. Each point $(x, y)$ becomes $(-x, y)$.

x	$f(x) = \sqrt{x}$	$g(x) = -\sqrt{x}$
0	0	0
1	1	-1
4	2	-2

x	$f(x) = \sqrt{x}$	$h(x) = \sqrt{-x}$
-4	undefined	2
-1	undefined	1
0	0	0
1	1	undefined
4	2	undefined

Symmetry

Symmetry with Respect to an Axis

Symmetry helps identify the geometric properties of graphs:

• y-axis symmetry: Replace $x$ with $-x$. If the equation is unchanged, the graph is symmetric about the y-axis.

• x-axis symmetry: Replace $y$ with $-y$. If the equation is unchanged, the graph is symmetric about the x-axis.

Symmetry with Respect to the Origin

A graph is symmetric with respect to the origin if replacing $x$ with $-x$ and $y$ with $-y$ yields an equivalent equation.

	x-axis	y-axis	Origin
Equation is unchanged if:	y is replaced with -y	x is replaced with -x	x is replaced with -x and y is replaced with -y
Example:

• A graph symmetric with respect to both axes is also symmetric with respect to the origin.

• A graph symmetric with respect to the origin need not be symmetric with respect to either axis.

• Possessing any two types of symmetry implies the third.

• A graph symmetric with respect to the x-axis does not represent a function.

Even and Odd Functions

Definitions

• Even Function: $f(-x) = f(x)$ for all $x$ in the domain. The graph is symmetric with respect to the y-axis.

• Odd Function: $f(-x) = -f(x)$ for all $x$ in the domain. The graph is symmetric with respect to the origin.

Example: $f(x) = x^2$ is even, $f(x) = x^3$ is odd, $f(x) = x + 1$ is neither.

Translations

Vertical Translations

Given $g(x) = f(x) + c$, the graph of $g$ is the graph of $f$ shifted vertically:

• Up: If $c > 0$, shift up by $c$ units.

• Down: If $c < 0$, shift down by $|c|$ units.

Horizontal Translations

Given $g(x) = f(x + c)$, the graph of $g$ is the graph of $f$ shifted horizontally:

• Left: If $c > 0$, shift left by $c$ units.

• Right: If $c < 0$, shift right by $|c|$ units.

Caution: For $f(x - h)$, the graph shifts right by $h$ units; for $f(x + h)$, it shifts left by $h$ units.

Summary of Graphing Techniques

• The graph of $f(x) + k$ is translated $k$ units up.

• The graph of $f(x) - k$ is translated $k$ units down.

• The graph of $f(x + h)$ is translated $h$ units to the left.

• The graph of $f(x - h)$ is translated $h$ units to the right.

• The graph of $a f(x)$ is a vertical stretch if $a > 1$, vertical shrink if $0 < a < 1$.

• The graph of $f(a x)$ is a horizontal stretch if $0 < a < 1$, horizontal shrink if $a > 1$.

• The graph of $-f(x)$ is reflected across the x-axis.

• The graph of $f(-x)$ is reflected across the y-axis.