Exponential Function

Exponential Functions Intro/Review

Multiple Choice

Which function is an exponential function?

• $$f\left(x\right)=2x$$  

• $$g\left(x\right)=x^2$$  

• $$h\left(x\right)=2^x$$  

Equations of Exponential Functions

*the base cannot be 1

Examples of Exp Functions:

g(x) = 8^x+2

h(x) = 0.5^x - 4

j(x)=​ 7(9)^x + 2

In an exponential function, there is a base* containing a variable in its exponent like f(x) = 2^x, where 2 is the base and x is the variable exponent.

NOT Exponential Functions

a(x) = x³ + 1

h(x) = 5x - 4

Multiple Choice

Does f(x) represent an exponential function?

$$f\left(x\right)=4x^3+5$$  

• Yes

• No

Multiple Choice

Does g(x) represent an exponential function?

$$g\left(x\right)=3^{x-1}$$  

• Yes

• No

Multiple Choice

Does h(x) represent an exponential function?

$$h\left(x\right)=5x+2$$  

• Yes

• No

Multiple Choice

In the exponential function f(x) = 5^x+2 - 1 , what is the base?

• 5

• x+2

• 2

• -1

Multiple Choice

In the exponential function f(x) = 2 + 7^{x - 4} , what is the base?

• 2

• 7

• x-4

• 4

Draw

Graph f(x)=2^x given its table of values.

Graphs of Exponential Functions

• is an increasing OR decreasing curve

• and approaches an asymptote* on one end ​

*an asymptote is a line that the graph APPROACHES but never intersects

f(x)=2^x

increasing

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g(x)= -2^x

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decreasing

​

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h(x)=2^x - 3

​

increasing

​

​

j(x)= (1/2)^x

​

decreasing

​

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Examples of Exponential Graphs

Multiple Choice

Which the following graphs represents an increasing exponential function?

Multiple Choice

Which of the following graphs depicts a decreasing exponential function?

In f(x) = 2^x , the asymptote is the x-axis. As an equation, the asymptote is y = 0​.

​

All exponential functions have HORIZONTAL asymptotes and can be written in the form

y = __ ​

​

​

Asymptotes - a line the graph APPROACHES but never intersects

asymptote: y = 4

​

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More Examples

asymptote: y = -3

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Multiple Choice

TRUE OR FALSE

The line shown on the graph is an asymptote for the exponential curve shown on the graph.

• TRUE

• FALSE

Multiple Choice

TRUE OR FALSE

The line shown on the graph is an asymptote for the exponential curve shown on the graph.

• TRUE

• FALSE

Multiple Choice

What is the equation for the horizontal asymptote?

• x = -4

• x = 7

• y = 6

• y = 7

Multiple Choice

TRUE OR FALSE

The line shown on the graph is an asymptote for the exponential curve shown on the graph.

• TRUE

• FALSE

Multiple Choice

What is the asymptote of the graph shown?

• $$y=0$$

• $$y=1$$

• $$y=2$$

• $$y=3$$

Multiple Choice

What is the x-intercept of the exponential function with the x-axis (y=0) as its asymptote? 

• (0,1)
• (0,0)
• There is no x-intercept

Multiple Choice

What is the domain of the exponential function with asymptote of y = -3 ?

• $$\left(-\infty,\infty\right)$$  

• $$\left(-2,\infty\right)$$  

• $$\left(-3,\infty\right)$$  

Multiple Choice

What is the range of the exponential function with asymptote of y = -3 ?

• $$\left(-\infty,\infty\right)$$  

• $$\left(-2,\infty\right)$$  

• $$\left(-3,\infty\right)$$  

Domain & Range

• Domain for all exponential functions is (-inf, inf) because

  • the graph is continuous forever to the left and right

  • any real number can be substituted ​into x in the exponent​

​

• Range has a constraint because of the asymptote

  • examples of possible ranges:

    • (-3 , inf)

    • (-inf, 4)

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Multiple Choice

What is the domain of the exponential function with asymptote of y = 0?

• $$\left(-\infty,\infty\right)$$  

• $$\left(0,\infty\right)$$  

• $$\left(1,\infty\right)$$  

• $$\left(-\infty,0\right)$$  

• $$\left(-\infty,1\right)$$  

Multiple Choice

What is the range of the exponential function with asymptote of y = 0?

• $$\left(-\infty,\infty\right)$$  

• $$\left(0,\infty\right)$$  

• $$\left(1,\infty\right)$$  

• $$\left(-\infty,0\right)$$  

• $$\left(-\infty,1\right)$$  

Multiple Choice

What is the domain of the exponential function with asymptote of y = 4?

• $$\left(-\infty,\infty\right)$$  

• $$\left(3,\infty\right)$$  

• $$\left(4,\infty\right)$$  

• $$\left(-\infty,3\right)$$  

• $$\left(-\infty,4\right)$$  

Multiple Choice

What is the range of the exponential function with asymptote of y = 4?

• $$\left(-\infty,\infty\right)$$  

• $$\left(3,\infty\right)$$  

• $$\left(4,\infty\right)$$  

• $$\left(-\infty,3\right)$$  

• $$\left(-\infty,4\right)$$  

Poll

Which one does not belong?

A B C D

Open Ended

Which did you pick and why?

Open Ended

What's one (or more) thing you learned?

Open Ended

What's one (or more) wonder/question you have?