Vertical & Horizontal Dilations (College Board AP® Precalculus): Study Guide

Vertical dilations

What are graph dilations?

• When you alter a function in certain ways, the effects on the graph of the function can be described by geometrical transformations

• With a dilation all the points on the graph are moved towards or away from either the x- or the y-axis by a constant scale factor

What is a vertical dilation?

• If you start with a function $f\left(x\right)$ and multiply the output by a constant $a$ (where $a\ne 0$)

  • the new function $g\left(x\right)=af\left(x\right)$ is called a multiplicative transformation of $f$

• The effect on the graph is a vertical dilation (stretch or compression)

  • every point on the graph is moved farther from or closer to the $\mathit{x}$-axis

    • If $\left|a\right|>1$, the graph is stretched vertically (pulled away from the $x$-axis)

    • If $0<\left|a\right|<1$, the graph is compressed vertically (pushed toward the $x$-axis)

  • If $a<0$, the graph is also reflected over the $x$-axis (i.e. flipped upside down around the $x$-axis), in addition to the stretch or compression

• The dilation factor is $\left|a\right|$

  • This tells you how much the graph is dilated (stretched or compressed)

  • Points on the graph after the dilation are $\left|a\right|$ times as far from the $x$-axis as they were before the dilation

• E.g. for a function $f\left(x\right)$

  • $g\left(x\right)=\frac{1}{3}f\left(x\right)$ dilates the graph vertically by a factor of $\frac{1}{3}$

    • this is a compression

    • the $y$-coordinates of all points are multiplied by $\frac{1}{3}$ (while $x$-coordinates stay the same)

  • $h\left(x\right)=-2f\left(x\right)$ dilates the graph by a factor of 2 and reflects it over the $x$-axis

    • this is a stretch (along with the reflection)

    • the $y$-coordinates of all points are multiplied by $-2$ (while $x$-coordinates stay the same)

How does a vertical dilation affect key features?

• Every output value is multiplied by $\mathit{a}$, so

  • $\mathit{y}$-intercept

    • multiplied by $a$

  • Local maxima and minima

    • their $y$-values are multiplied by $a$

      • and if $a<0$, maxima become minima and vice versa

  • Horizontal asymptotes

    • the $y$-value defining the asymptote is multiplied by $a$

• Features that depend only on $\mathit{x}$-values are not affected

  • Zeros ($x$-intercepts) stay in the same place

    • since $a·0=0$

  • Vertical asymptotes stay in the same place

  • Intervals of increase/decrease stay the same

    • unless $a<0$, which reverses them

How do I find the analytical form of a vertical dilation?

• $af\left(x\right)$ means 'multiply the entire function $\mathit{f}$ by $\mathit{a}$'

  • So if you have an expression for $f\left(x\right)$, simply multiply it by $a$ to get the analytical form of the transformed function

  • Simplify if necessary

• E.g. if $f\left(x\right)=x^2-3x+4$

  • Then to find $g\left(x\right)=2f\left(x\right)$

    • $g\left(x\right)=2\left(x^2-3x+4\right)=2x^2-6x+8$

Horizontal dilations

What is a horizontal dilation?

• If you start with a function $f\left(x\right)$ and multiply the input by a constant $\mathit{b}$ (where $b\ne 0$) before the function is applied

  • the new function $g\left(x\right)=f\left(bx\right)$ is also called a multiplicative transformation of $f$

    • i.e., vertical and horizontal dilations are both types of multiplicative transformation

• The effect on the graph is a horizontal dilation (stretch or compression)

  • every point on the graph is moved farther from or closer to the $\mathit{y}$-axis

    • If $\left|b\right|>1$, the graph is compressed horizontally (pushed toward the $y$-axis) by a factor of $\frac{1}{\left|b\right|}$

    • If $0<\left|b\right|<1$, the graph is stretched horizontally (pulled away from the $y$-axis) by a factor of $\frac{1}{\left|b\right|}$

  • If $b<0$, the graph is also reflected over the $y$-axis (i.e. flipped left-to-right around the $y$-axis), in addition to the stretch or compression

• The dilation factor is $\frac{1}{\left|b\right|}$

  • This tells you how much the graph is dilated (stretched or compressed)

  • Points on the graph after the dilation are $\frac{1}{\left|b\right|}$ times as far from the $y$-axis as they were before the dilation

• Note the counterintuitive behavior

  • the dilation factor is $\frac{1}{\left|b\right|}$, not $\left|b\right|$

    • Multiplying the input by a number greater than 1 actually compresses the graph, rather than stretching it

• E.g. for a function $f\left(x\right)$

  • $g\left(x\right)=f\left(2x\right)$ dilates the graph horizontally by a factor of $\frac{1}{2}$

    • this is a compression

    • the $x$-coordinates of all points are multiplied by $\frac{1}{2}$ (while $y$-coordinates stay the same)

  • $h\left(x\right)=f\left(-\frac{1}{3}x\right)$ dilates the graph horizontally by a factor of 3 and reflects it over the $y$-axis

    • this is a stretch (along with the reflection)

    • the $x$-coordinates of all points are multiplied by $-3$ (while $y$-coordinates stay the same)

How does a horizontal dilation affect key features?

• Every $\mathit{x}$-value on the graph is multiplied by $\frac{1}{b}$, so features that depend on $x$-values are affected:

  • Zeros ($x$-intercepts)

    • their $x$-values are multiplied by $\frac{1}{b}$

  • Local maxima and minima

    • their $x$-values are multiplied by $\frac{1}{b}$, but the $y$-values stay the same

  • Vertical asymptotes

    • their $x$-values are multiplied by $\frac{1}{b}$

• Features that depend only on $\mathit{y}$-values are not affected

  • $\mathit{y}$-intercept stays the same

    • since $f(b·0)=f\left(0\right)$

  • Horizontal asymptotes stay at the same $y$-value

How do I find the analytical form of a horizontal dilation?

• $f\left(bx\right)$ means 'replace $\mathit{x}$ everywhere in $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}$with $\mathit{b}\mathit{x}$'

  • So if you have an expression for $f\left(x\right)$, simply put $bx$ everywhere that $x$is to get the analytical form of the transformed function

  • Simplify if necessary

• E.g. if $f\left(x\right)=x^2-3x+4$

  • Then to find $g\left(x\right)=f\left(2x\right)$

    • $g\left(x\right)={\left(2x\right)}^2-3\left(2x\right)+4=4x^2-6x+4$

How can you tell a vertical and horizontal dilation apart?

• Vertical dilation $g\left(x\right)=af\left(x\right)$

  • the constant $a$ multiplies the output

    • it appears outside the function

• Horizontal dilation $g\left(x\right)=f\left(bx\right)$

  • the constant $b$ multiplies the input

    • it appears inside the function

• A common mistake is to confuse the two

  • Just like with translations, the key question is:

    • Is the constant being applied to the input or the output?