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

Vertical translations

What are graph translations?

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

• With a translation the shape, size, and orientation of the graph remain unchanged

  • the graph is merely shifted (up or down, left or right) in the xy plane

What is a vertical translation?

• If you start with a function $f\left(x\right)$ and add a constant $\mathit{k}$ to the output

  • the new function $g\left(x\right)=f\left(x\right)+k$ is called an additive transformation of $f$

• The effect on the graph is a vertical translation (shift)

  • every point on the graph moves up or down by $k$ units

    • If $k>0$, the graph shifts up by $k$ units

    • If $k<0$, the graph shifts down by $\left|k\right|$ units

• The shape of the graph does not change

  • it is simply moved vertically

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

  • $g\left(x\right)=f\left(x\right)+1$ shifts the entire graph of $f$ up by 1 unit

  • $h\left(x\right)=f\left(x\right)-4$ shifts it down by 4 units

How does a vertical translation affect key features?

• Every output value changes by $\mathit{k}$, so

  • $\mathit{y}$-intercept

    • changes by $k$

  • Local maxima and minima

    • their $y$-values shift by $k$

    • but the $x$-values where they occur stay the same

  • Horizontal asymptotes

    • shift by $k$

      • e.g. if the original asymptote is $y=0$, the new one is $y=k$

  • Zeros ($x$-intercepts)

    • generally change, because the outputs for each $x$ value are different

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

  • Vertical asymptotes stay in the same place

  • Intervals of increase/decrease stay the same

  • Concavity intervals (and the concavity on them) stay the same

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

• $f\left(x\right)+k$ means 'add $\mathit{k}$ to the entire function'

  • So if you have an expression for $f\left(x\right)$, simply add $k$ to it 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 $g\left(x\right)=f\left(x\right)+2$

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

Horizontal translations

What is a horizontal translation?

• If you start with a function $f\left(x\right)$ and add a constant $\mathit{h}$ to the input before the function is applied

  • the new function $g\left(x\right)=f(x+h)$ is also called an additive transformation of $f$

    • i.e., vertical and horizontal translations are both types of additive transformation

• The effect on the graph is a horizontal translation (shift)

  • every point on the graph moves left or right

    • If $h>0$, the graph shifts to the left by $h$ units

    • If $h<0$, the graph shifts to the right by $\left|h\right|$ units

  • Note the counterintuitive direction

    • adding a positive number inside the function moves the graph to the left, not the right

• The shape of the graph does not change

  • it is simply moved horizontally

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

  • $g\left(x\right)=f\left(x+3\right)$ shifts the entire graph of $f$ left by 3 units

  • $h\left(x\right)=f(x-2)$ shifts it right by 2 units

How does a horizontal translation affect key features?

• Every input value is shifted, so features that depend on $x$-values move:

  • Zeros ($x$-intercepts)

    • shift left or right by the same amount as the graph

  • Local maxima and minima

    • their $x$-values shift, but the $y$-values stay the same

  • Vertical asymptotes

    • shift horizontally the same way as the rest of the graph

    • e.g. if the original asymptote is $x=0$, the new one is $x=-h$

  • Intervals of increase/decrease

    • shift horizontally (same pattern, different $x$-values)

  • Concavity intervals

    • shift horizontally (same pattern, different $x$-values)

    • though the concavity on those intervals stays the same

• Features that depend only on the overall shape or on the $\mathit{y}$-values are not affected:

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

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

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

  • So if you have an expression for $f\left(x\right)$, simply put $x+h$ 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 $g\left(x\right)=f\left(x+2\right)$

    • $\begin{array}{rcl}g\left(x\right)& =& {\left(x+2\right)}^2-3\left(x+2\right)+4=x^2+4x+4-3x-6+4=x^2+x+2\end{array}$

How can you tell vertical and horizontal translations apart?

• Vertical translation $g\left(x\right)=f\left(x\right)+k$

  • the constant $k$ is added outside the function

    • it changes the output

• Horizontal translation $g\left(x\right)=f(x+h)$

  • the constant $h$ is added inside the function (to the input)

    • it changes which input produces each output

• A common mistake is to confuse the two

  • If you see a constant being added to $f\left(x\right)$, that's vertical

  • If you see a constant being added to $x$ inside $f$, that's horizontal