Exponential Transformations and Characteristics

The Exponential Functions: Attributes and Transformations

The Basics

An exponential function with base b is defined by $f\left(x\right)=ab^x$ where a ≠0, b > 0 , b ≠1, and x is any real number.

The base, b, is constant and the exponent, x, is a variable.

The Asymptote of Exponential Function

• Asymptote - The line that the function approaches but never touches

  The exponential function has a horizontal asymptote

• For the function f(x) =  $2^x$                                                                       its asymptote is the x-axis or y=0 

Multiple Choice

Which equation best describes the graph?

• y = 2^x + 4

• y = 2^x - 3

• y = 2^x - 4

• y = 2^x + 3

Multiple Choice

Find the domain.

• all values less than 1

• all real numbers

• all positive numbers

• all values less than 0

Multiple Choice

Determine the asymptote for the function

f(x)=3(2)^x-4+5

• x=4

• y=4

• x=5

• y=5

Multiple Choice

What is the horizontal asymptote for the graph of

f(x)=-2(3)^x+4-8 ?

• y=8

• y=-8

• x=-8

• x=-4

Multiple Choice

Asymptote?

• y = 2

• y = -2

• x = 2

• y = 0

Multiple Choice

What is the horizontal asymptote?

• y = 2

• y = -2

• x = 2

• y = 0

Multiple Choice

What is the horizontal asymptote?

• y=3

• y=-2

• y=-3

• (0,-3)

Multiple Choice

What is the range of the function shown below?

• 0 < y < 12

• -2 < y < 5

• y > 0

• All Real Numbers

Multiple Choice

What is the domain of the function shown below?

• -2 < x < 5

• 0 < x < 12

• y > 0

• All Real Numbers

Multiple Choice

What is the range of this graph?

• y = 6

• (0, -4)

• y > -6

• y < -6

Multiple Choice

What is the asymptote value in the function?

• 3

• 2

• 7

• 0

Multiple Choice

What is the y-intercept of the function?

• 2
• 3
• 1
• -2

Multiple Choice

What is the transformation?

• Horizontal Translation left 2

• Vertical Translation down 2

• None

• Reflection across the x-axis

Multiple Choice

Which of the following value translates the parent function down?

• 0

• 1

• -1

• -2

Multiple Choice

What is the transformation?

• Vertical Translation up 2 and Horizontal translation right 1

• Vertical Translation down 2 and Horizontal Translation left 1

• Horizontal Translation left 2 and Vertical Translation up 1

• Horizontal Translation right 2 and Vertical Translation down 1

Multiple Choice

What is the transformation?

• Vertical Translation up 2

• Vertical Translation down 2

• Horizontal Translation left 2

• Horizontal Translation right 2

Multiple Choice

What is the transformation?

• Vertical Translation up 1

• Vertical Translation down 1

• Horizontal Translation left 1

• Horizontal Translation right 1

Multiple Choice

Identify the y-intercept of the function $$-3\left(3\right)^x+4$$  

• y-intercept at (0, -3)

• y-intercept at (0, 1)

• y-intercept at (0, -7)

Multiple Choice

Identify the asymptote of the function $$-3\left(3\right)^x+4$$  

• y = -3

• y = 1

• y = 4

Multiple Choice

How has the function $$7\left(3\right)^x-5$$ been transformed from the parent function  $$7\left(3\right)^x$$ ?

• Shifted up 5

• Shifted down 5

• Reflection over the x-axis

• Reflection over the y-axis

Multiple Choice

What is the y-intercept of the function $$y=4\left(5\right)^x?$$  

• (0, 0)

• (0, 4)

• (0, 5)

Multiple Choice

What is the asymptote of the function $$y=4\left(5\right)^x?$$  

• y = 0

• y = 4

• y = 5

Multiple Choice

 If f(x) is a transformation of g(x), describe the transformation.

$$g\left(x\right)=0.5^{x-3}$$   
$$f\left(x\right)=0.5^x$$  
Be careful!  This is not the same question at #4!

• Left 3 units

• Right 3 units

• Up 3 units

• Down 3 units

Multiple Choice

 If g(x) is a transformation of f(x), describe the transformation.

$$g\left(x\right)=0.5^{x-3}$$   
$$f\left(x\right)=0.5^x$$  

• Left 3 units

• Right 3 units

• Up 3 units

• Down 3 units

Multiple Choice

 If g(x) is a transformation of f(x), describe the transformation.

$$g\left(x\right)=\left(-3\right)2^x$$   

$$f\left(x\right)=\left(3\right)2^x$$  

• Down 3 units

• Stretch vertically

• Horizontal reflection

• Vertical reflection

Multiple Choice

 If g(x) is a transformation of f(x), describe the transformation.

$$f\left(x\right)=0.4^x-1$$
$$g\left(x\right)=0.4^x-2$$  

 

• Down 1 unit
  

• Down 2 units

• Up 1 unit

• Up 2 units

Multiple Choice

If g(x) is a transformation of f(x), describe the transformation.

$$f\left(x\right)=2^x$$      $$g\left(x\right)=2^x+4$$   

• Up 4 units

• Down 4 units

• Left 4 units

• Right 4 units