An exponential relationship can be modeled by a function with the independent variable in the exponent, known as an **exponential function**:

\displaystyle f\left(x\right)=ab^x

\bm{a}

Leading coefficient

\bm{b}

Base where b>0, b\neq 1

\bm{x}

Independent variable

\bm{f\left(x\right)}

Dependent variable

We can also call b the growth or decay factor.

Move the sliders for a and b to see how the base and the exponent affect the graph of the exponential function.

- What happens to the graph as b increases?
- What happens to the graph as b is between 0 and 1?
- What happens to the graph as a decreases?
- Describe the end behavior for:
- a>0 and b>1?
- a>0 and 0<b<1?
- a<0 and b>1?
- a<0 and 0<b<1?

- Create two equations where the function is increasing over its domain.
- Create two equations where the function is decreasing over its domain.

The general shape for different values of a and b is shown.

y>0, Increases at an increasing rate

y>0, Decreases at a decreasing rate

y<0, Decreases at an increasing rate

y<0, Increases at a decreasing rate

All exponential functions of the form y=ab^x have the following features in common:

- The domain is -\infty <x<\infty.
- The y-intercept is at \left(0,a\right).
- There is a horizontal asymptote at y=0.

Which graph(s) show an exponential function? Select all correct graphs.

A

B

C

D

Worked Solution

Consider the two functions, f\left(x\right) = 2 \left(\dfrac{1}{3}\right)^x and g\left(x\right) = x^2+2.

a

Complete the table of values and sketch f\left(x\right) = 2 \left(\dfrac{1}{3}\right)^x.

x | -2 | -1 | 0 | 1 | 2 |
---|---|---|---|---|---|

y |

Worked Solution

b

Sketch the graph of g\left(x\right) on the same plane as f\left(x\right).

Worked Solution

c

Compare the y-intercepts of both functions.

Worked Solution

d

Compare the end behavior of both functions.

Worked Solution

e

Compare the domain and range of both functions.

Worked Solution

Idea summary

Exponential functions can take the form:

\displaystyle f\left(x\right)=ab^x

\bm{a}

Leading coefficient

\bm{b}

Base where b>0, b\neq 1

\bm{x}

Independent variable

\bm{f\left(x\right)}

Dependent variable

Exponential functions of the form y=ab^x have the following features in common:

- The domain is -\infty <x<\infty.
- The y-intercept is at (0,a).
- There is a horizontal asymptote at y=0.

To draw the graph of an **exponential function**, we can use a variety of strategies, including:

- Completing a table of values for the function and drawing the curve through the points found
- Using transformations
- Identifying key features from the equation
- Using technology, such as a physical or online graphing calculator

Move the sliders to see how each one affects the graph.

- What happens to the graph when k is positive or negative?
- What happens to the graph when h is positive or negative?
- What happens to the graph when a is positive or negative?
- What happens to the graph when c is positive or negative?
- What impact does c have on the graph?
- What happens to the graph when 0<b<1?

For the function:

\displaystyle f\left(x\right)=a\cdot b^{c\left(x-h\right)}+k

\bm{b}

is the base

\bm{a}

can reflect, stretch or compress vertically

\bm{c}

can reflect, stretch or compress horizontally

\bm{h}

determines the horizontal translation

\bm{k}

determines the vertical translation

\text{Reflection across the }x\text{-axis:} | y=-b^x |
---|---|

\text{Reflection across the }y\text{-axis:} | y=b^{-x} |

\text{Vertical stretch when} \left|a\right|>1 \\ \text{Vertical compression when } 0<\left|a\right|<1 \text{:} | y=a\left(b\right)^x |

\text{Horizontal compression when} \left|c\right|>1 \\ \text{Horizontal stretch when} 0<\left|c\right|<1 \text{:} | y=b^{c\cdot x} |

\text{Horizontal translation by } h \\ \text{Vertical translation by } k \text{:} | y=b^{x-h}+k |

For horizontal stretches and compressions, recall c=\dfrac{1}{\text{scale factor}}

The parent function y=b^x has key features:

- Increasing for b>1, and decreasing for 0<b<1
- Domain of \left(-\infty, \infty\right) and range of \left[0, \infty\right)
- y=0 is the horizontal asymptote
- y-intercept at \left(0,1\right)
- Another point at \left(1,b\right)

To get a more precise shape, we can map the asymptote and the points \left(0,1\right) and \left(1,b\right) using the transformations.

For each graph, first identify the transformation from the parent function f\left(x\right)=\left(\dfrac{1}{3}\right)^x, then write the equation of g\left(x\right).

a

Worked Solution

b

Worked Solution

c

Worked Solution

Given the parent function f\left(x\right) = 4^x, graph g\left(x\right).

a

g\left(x\right) = 4^{-x}

Worked Solution

b

g\left(x\right) = -4^{x}+3

Worked Solution

c

g\left(x\right) = - \left(\dfrac{1}{4}\right)\cdot 4^x

Worked Solution

Consider the following exponential functions:

x | -2 | -1 | 0 | 1 | 2 |
---|---|---|---|---|---|

g\left(x\right) | -9 | -3 | -1 | -\dfrac{1}{3} | -\dfrac{1}{9} |

a

Determine which function increases at a slower rate.

Worked Solution

b

Identify the y-intercept for each function.

Worked Solution

c

Identify and compare f(-1) and g(-1).

Worked Solution

Idea summary

We can use transformations to graph an exponential function in the form {y=a \cdot b^{c (x-h)} + k} and identify key features:

\text{Reflection across the }x\text{-axis:} | y=-b^x |
---|---|

\text{Reflection across the }y\text{-axis:} | y=b^{-x} |

\text{Vertical stretch when} \left|a\right|>1 \\ \text{Vertical compression when } 0<\left|a\right|<1 \text{:} | y=a\left(b\right)^x |

\text{Horizontal compression when} \left|c\right|>1 \\ \text{Horizontal stretch when} 0<\left|c\right|<1 \text{:} | y=b^{c\cdot x} |

\text{Horizontal translation by } h \\ \text{Vertical translation by } k \text{:} | y=b^{x-h}+k |

For horizontal stretches and compressions, recall c=\dfrac{1}{\text{scale factor}}

The parent function y=b^x has key features:

- Increasing for b>1, and decreasing for 0<b<1
- Domain of \left(-\infty, \infty\right) and range of \left[0, \infty\right)
- y=0 is the horizontal asymptote
- y-intercept at \left(0,1\right)
- Another point at \left(1,b\right)

Recall that when inverse functions are graphed on the same Cartesian plane, they are reflections of each other about the line y=x.

Each point \left(x,y\right) on the graph of f\left(x\right) corresponds to the point \left(y,x\right) on f^{-1}\left(x\right). The key features are affected in a similar way.

The y-intercept of f\left(x\right) becomes the x-intercept of f^{-1}\left(x\right).

The x-intercept of f\left(x\right) becomes the y-intercept of f^{-1}\left(x\right).

The horizontal asymptote of f\left(x\right), y=a, becomes the vertical asymptote of f^{-1}\left(x\right), x=a.

The domain of f\left(x\right) becomes the range of f^{-1}\left(x\right).

The range of f\left(x\right) becomes the domain of f^{-1}\left(x\right).

Consider the graph of 3^x.

a

Sketch the inverse function on the same coordinate plane.

Worked Solution