# Horizontal and Vertical Translations of Exponential Functions

### Learning Objectives

- Graph exponential functions shifted horizontally or vertically and write the associated equation

## Graphing a Vertical Shift

The first transformation occurs when we add a constant*d*to the parent function [latex]f\left(x\right)={b}^{x}[/latex], giving us a

**vertical shift**

*d*units in the same direction as the sign. For example, if we begin by graphing a parent function, [latex]f\left(x\right)={2}^{x}[/latex], we can then graph two vertical shifts alongside it, using [latex]d=3[/latex]: the upward shift, [latex]g\left(x\right)={2}^{x}+3[/latex] and the downward shift, [latex]h\left(x\right)={2}^{x}-3[/latex]. Both vertical shifts are shown in the figure below. Observe the results of shifting [latex]f\left(x\right)={2}^{x}[/latex] vertically:

- The domain, [latex]\left(-\infty ,\infty \right)[/latex] remains unchanged.
- When the function is shifted up 3 units to [latex]g\left(x\right)={2}^{x}+3[/latex]:
- The
*y-*intercept shifts up 3 units to [latex]\left(0,4\right)[/latex]. - The asymptote shifts up 3 units to [latex]y=3[/latex].
- The range becomes [latex]\left(3,\infty \right)[/latex].

- The
- When the function is shifted down 3 units to [latex]h\left(x\right)={2}^{x}-3[/latex]:
- The
*y-*intercept shifts down 3 units to [latex]\left(0,-2\right)[/latex]. - The asymptote also shifts down 3 units to [latex]y=-3[/latex].
- The range becomes [latex]\left(-3,\infty \right)[/latex].

- The

### Try it

- Use the slider in the graph below to create a graph of [latex]f(x) = 2^x[/latex] that has been shifted 4 units up. Add a line that represents the horizontal asymptote for this function. What is the equation for this function? What is the new y-intercept? What is it's domain and range?
- Now create a graph of the function [latex]f(x) = 2^x[/latex] that has been shifted down 2 units. Add a line that represents the horizontal asymptote.What is the equation for this function? What is the new y-intercept? What is it's domain and range?

Answer:

- Equation: [latex]f(x) = 2^x+4[/latex], Horizontal Asymptote: [latex]y = 4[/latex], y-intercept: [latex](0,5)[/latex]Domain: [latex](-\infty,\infty)[/latex], Range: [latex](4,\infty)[/latex]
- Equation: [latex]f(x) = 2^x-2[/latex], Horizontal Asymptote: [latex]y = -2[/latex], y-intercept: [latex](0,-1)[/latex]Domain: [latex](-\infty,\infty)[/latex], Range: [latex](-2,\infty)[/latex]

## Graphing a Horizontal Shift

The next transformation occurs when we add a constant*c*to the input of the parent function [latex]f\left(x\right)={b}^{x}[/latex], giving us a

**horizontal shift**

*c*units in the

*opposite*direction of the sign. For example, if we begin by graphing the parent function [latex]f\left(x\right)={2}^{x}[/latex], we can then graph two horizontal shifts alongside it, using [latex]c=3[/latex]: the shift left, [latex]g\left(x\right)={2}^{x+3}[/latex], and the shift right, [latex]h\left(x\right)={2}^{x - 3}[/latex]. Both horizontal shifts are shown in the figure below. Observe the results of shifting [latex]f\left(x\right)={2}^{x}[/latex] horizontally:

- The domain, [latex]\left(-\infty ,\infty \right)[/latex], remains unchanged.
- The asymptote, [latex]y=0[/latex], remains unchanged.
- The
*y-*intercept shifts such that:- When the function is shifted left 3 units to [latex]g\left(x\right)={2}^{x+3}[/latex], the
*y*-intercept becomes [latex]\left(0,8\right)[/latex]. This is because [latex]{2}^{x+3}=\left(8\right){2}^{x}[/latex], so the initial value of the function is 8. - When the function is shifted right 3 units to [latex]h\left(x\right)={2}^{x - 3}[/latex], the
*y*-intercept becomes [latex]\left(0,\frac{1}{8}\right)[/latex]. Again, see that [latex]{2}^{x - 3}=\left(\frac{1}{8}\right){2}^{x}[/latex], so the initial value of the function is [latex]\frac{1}{8}[/latex].

- When the function is shifted left 3 units to [latex]g\left(x\right)={2}^{x+3}[/latex], the

### try it

- Use the slider in the graph below to create a graph of [latex]f(x) = 2^x[/latex] that has been shifted 4 units to the right. What is the equation for this function? What is the new y-intercept? What are it's domain and range?
- Use the slider in the graph below to create a graph of [latex]f(x) = 2^x[/latex] that has been shifted 3 units to the left. What is the equation for this function? What is the new y-intercept? What are it's domain and range?

Answer:

- Equation: [latex]f(x) = 2^{x-4}[/latex], y-intercept: [latex](0,\frac{1}{32})[/latex]Domain: [latex](-\infty,\infty)[/latex], Range: [latex](0,\infty)[/latex]
- Equation: [latex]f(x) = 2^{x+3}[/latex], y-intercept: [latex](0,8)[/latex]Domain: [latex](-\infty,\infty)[/latex], Range: [latex](0,\infty)[/latex]

### A General Note: Shifts of the Parent Function [latex]f\left(x\right)={b}^{x}[/latex]

For any constants*c*and

*d*, the function [latex]f\left(x\right)={b}^{x+c}+d[/latex] shifts the parent function [latex]f\left(x\right)={b}^{x}[/latex]

- vertically
*d*units, in the*same*direction of the sign of*d*. - horizontally
*c*units, in the*opposite*direction of the sign of*c*. - The
*y*-intercept becomes [latex]\left(0,{b}^{c}+d\right)[/latex]. - The horizontal asymptote becomes
*y*=*d*. - The range becomes [latex]\left(d,\infty \right)[/latex].
- The domain, [latex]\left(-\infty ,\infty \right)[/latex], remains unchanged.

### How To: Given an exponential function with the form [latex]f\left(x\right)={b}^{x+c}+d[/latex], graph the translation.

- Draw the horizontal asymptote
*y*=*d*. - Identify the shift as [latex]\left(-c,d\right)[/latex]. Shift the graph of [latex]f\left(x\right)={b}^{x}[/latex] left
*c*units if*c*is positive, and right [latex]c[/latex] units if*c*is negative. - Shift the graph of [latex]f\left(x\right)={b}^{x}[/latex] up
*d*units if*d*is positive, and down*d*units if*d*is negative. - State the domain, [latex]\left(-\infty ,\infty \right)[/latex], the range, [latex]\left(d,\infty \right)[/latex], and the horizontal asymptote [latex]y=d[/latex].

### Example: Graphing a Shift of an Exponential Function

Graph [latex]f\left(x\right)={2}^{x+1}-3[/latex]. State the domain, range, and asymptote.Answer: We have an exponential equation of the form [latex]f\left(x\right)={b}^{x+c}+d[/latex], with [latex]b=2[/latex], [latex]c=1[/latex], and [latex]d=-3[/latex]. Draw the horizontal asymptote [latex]y=d[/latex], so draw [latex]y=-3[/latex]. Identify the shift as [latex]\left(-c,d\right)[/latex], so the shift is [latex]\left(-1,-3\right)[/latex].

### Try It

Use the sliders in the graph below to create a graph of the function [latex]f\left(x\right)={2}^{x - 1}+3[/latex]. State domain, range, and asymptote. https://www.desmos.com/calculator/e5l4eca3obAnswer:
The domain is [latex]\left(-\infty ,\infty \right)[/latex]; the range is [latex]\left(3,\infty \right)[/latex]; the horizontal asymptote is *y *= 3.

## Use a Graph to Approximate a Solution to an Exponential Equation

Graphing can help you confirm or find the solution to an exponential equation. An exponential equation is different from a function because a function is a large collection of points made of inputs and corresponding outputs, whereas equations that you have seen typically have one, two, or no solutions. For example, [latex]f(x)=2^{x}[/latex] is a function and is comprised of many points [latex](x,f(x))[/latex], and [latex]4=2^{x}[/latex] can be solved to find the specific value for x that makes it a true statement. The graph below shows the intersection of the line [latex]f(x)=4[/latex], and [latex]f(x)=2^{x}[/latex], you can see they cross at [latex]y=4[/latex]. https://www.desmos.com/calculator/lhmpdkbjt0 In the next example, you can try this for yourself.### Example : Approximating the Solution of an Exponential Equation

Use Desmos to solve [latex]42=1.2{\left(5\right)}^{x}+2.8[/latex] graphically.Answer: First, graph the function [latex]f(x)=1.2{\left(5\right)}^{x}+2.8[/latex] , then add another function [latex]f(x) = 42[/latex]. Desmos automatically calculates points of interest including intersections. Essentially, you are looking for the intersection of two functions. Click on the point of intersection, and you will see the the x and y values for the point. Your graph will look like this: https://www.desmos.com/calculator/ozaejvejqn

### Try It

Solve [latex]4=7.85{\left(1.15\right)}^{x}-2.27[/latex] graphically. Round to the nearest thousandth.Answer: [latex]x\approx -1.608[/latex]

## Licenses & Attributions

### CC licensed content, Original

- Revision and Adaptation.
**Provided by:**Lumen Learning**License:**CC BY: Attribution. - Horizontal and Vertical Translations of Exponential Functions Interactive.
**Authored by:**Lumen Learning.**Located at:**https://www.desmos.com/calculator/5mrjqegkxk.**License:**Public Domain: No Known Copyright. - Horizontal and Vertical Translations of Exponential Functions 2 Interactive.
**Authored by:**Lumen Learning.**Located at:**https://www.desmos.com/calculator/rpv1kea0pz.**License:**Public Domain: No Known Copyright. - Horizontal and Vertical Translations of Exponential Functions 3 Interactive.
**Authored by:**Lumen Learning.**Located at:**https://www.desmos.com/calculator/e5l4eca3ob.**License:**Public Domain: No Known Copyright. - Solve Exponential Interactive.
**Authored by:**Lumen Learning.**Located at:**https://www.desmos.com/calculator/lhmpdkbjt0.**License:**Public Domain: No Known Copyright. - Horizontal and Vertical Translations of Exponential Functions 4 Interactive.
**Authored by:**Lumen Learning.**Located at:**https://www.desmos.com/calculator/ozaejvejqn.**License:**Public Domain: No Known Copyright.

### CC licensed content, Shared previously

- College Algebra.
**Provided by:**OpenStax**Authored by:**Abramson, Jay et al..**Located at:**https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites.**License:**CC BY: Attribution.**License terms:**Download for free at http://cnx.org/contents/[email protected]. - Question ID 63064.
**Authored by:**Brin,Leon.**License:**CC BY: Attribution.**License terms:**IMathAS Community License CC-BY + GPL. - Ex: Match the Graphs of Translated Exponential Function to Equations.
**Authored by:**James Sousa (Mathispower4u.com).**License:**CC BY: Attribution.