Exponential Functions

Vocabulary

100

an expert in or a student of mathematics.

What is a Mathematician?

200

The ratio between any two consecutive terms in a geometric sequence

2,4,8,16,... = 2

What is a Common Ratio?

300

The number of times the base is used as a factor

                               2³

What is the Exponent?

400

f(x) = a . b^x 

What is an Exponential Function?

500

A horizontal line that a function gets closer and closer to, but never intersects.

What is a Horizontal Asymptote?

True or False

100

2³ = 2 x 3 = 6

False

2³ = 2 x 2 x 2 = 8

200

In f(x) = 3 . 2^x

^{3 is the y-intercept}

TRUE

f(x) = a . b

a= y-intercept

b= common ratio

300

This graph is Increasing

TRUE

We read graphs from the left to the right

400

The h-value produces a horizontal translation (left/right) while the k-value produces a vertical translation (up/down)

TRUE

f(x) = a . b ^(x-h) + k

500

This table matches this function. f(x) = 3(1/2)^x

X    Y

-1    6

0     3

1     1.5

2     0.75

TRUE

Exponentials

100

What are the a & b values of this exponential function?

f(x) = 1 . 3^x

a = 1 (the y-intercept)

b = 3 (the common ratio)

200

What is the horizontal asymptote of this graph?

 

The horizontal asymptote is 3

y = 3

300

Evaluate the function.

f(x) = 1/3 . 6^x  at x = 2

f(2) = 1/3 . 6²

^{f(2) = 1/3 . 36}

^{f(2) = 36/3}

^{f(2) = 12}

400

Make a table of values for function:

f(x) = 4 . 1/2^x

X      Y

-1

0

1

2

f(x) = 4 . 1/2^x 

X       Y

-1       8

0        4

1        2

2        1

500

What are the y values of this function?

f(x) = 3 . (1/2)^x

^{X      Y}

^-2

^-1

⁰

¹

²

X     Y

-2     12

-1     6

0      3

1      1.5

2      0.75

What is ___?

100

What is the y-intercept of this table?

6

On a table, you find the y-intercept across from the 0 in the x column

200

a = y-intercept

b = common ratio

300

f(x) = 2 . 4^(x-3) + 5

Identify the values of a, b, h, & k

f(x) = a . b^(x-h) +k             f(x) = 2 . 4^(x-3) + 5

a=2

b=4

h=3

k=5

400

What is an equation that represents the function?

f(x) = 20 . (1/2)^x

500

What is the function that would translate f(x) four space UP

                       f(x) = 1 . 2^x

f(x) = 1 . 2^x + 4

Transforming exponential functions

100

Translate the function f(x) = 1 . 3^x to move 3 spaces to the right

f(x) = 1 . 3^(x-3)

200

What is the function f(x)? Translate the function 3 spaces to the right and up 2. What is the new function g(x)?

f(x) = 1 . 3^x

g(x) = 1 . 3^(x-3) + 2

300

Translate y = 10^x to move 2 units to the right. Also translate the function to move 2 units down.

y=10^x-2 will translate it 2 units to the right

y=10^x -2 will translate it 2 units down

400

Translate the function f(x) = 2^x to reflect over the y-axis and over the x-axis.

y = 2^-x reflects over the y=axis

y = -2^x reflects over the x-axis

500

Make a table for g(x) = 3 . 2^x +1

X     Y

-1

0

1

2

Then translate g(x) = 3 . 2^x  

to move to the left 2 and down 2

X     Y

-1     5/2

0      4

1      7

2      13

g(x) = 3 . 2^(x+2) -2