Question

For the function $g(x) =\frac{-28}{2^{5x+5} } + 7$


a) State the parent function in the form and express the function in the form
α · $(B^{k(x-d)}) + c$


b) State the transformations of g(x) in proper order from the parent function.


c) Express the function in the form $y = ab^{x} + c$


d) Determine the any asymptotes and state whether the function is an example of exponential growth or decay


e) Determine the domain and range of the function.


f) Calculate the x-intercept and y-intercept, then sketch the function.

Answer 1

The solutions to the questions are:

• The equation of the function is $g(x) = -\frac 78(2^{-5x})+ 7$
• The domain is -∝ < x < ∝ while the range is y < 7
• The horizontal asymptote is y = 7 and the function is an example of exponential decay
• The y-intercept is 49/8 while the x-intercept is -0.6

State the parent function in the form and express the function in the form

The function is an exponential function.

So, the parent function has the form

y = ab^x

Using the function in (a), the parent function is: y = -7/8 * 2^x and the form of the function is $g(x) = -\frac 78(2^{-5x})+ 7$

The transformation

First, the function is horizontally stretched by -5

This gives

$g(x) = -\frac 78(2^{-5x})$

First, the function is shifter up by 7 units

This gives

$g(x) = -\frac 78(2^{-5x})+ 7$

Express the function in the form y = ab^x + c

The equation of the function is given as:

g(x) = -28/[2^(5x + 5)] + 7

Rewrite the equation as follows:

g(x) = -28/[2^(5x) * 2^5] + 7

Evaluate the exponent

g(x) = -28/[2^(5x) * 32] + 7

Divide

g(x) = -7/[2^(5x) * 8] + 7

Rewrite as:

g(x) = -7/[8 * 2^(5x)] + 7

Further, rewrite as:

g(x) = -7/8 * 2^(-5x) + 7

Rewrite properly as:

$g(x) = -\frac 78* 2^{-5x} + 7$

Determine any asymptotes and state whether the function is an example of exponential growth or decay

We have:

g(x) = -7/8 * 2^(-5x) + 7

Set the radical to 0

g(x) = 0 + 7

Evaluate

g(x) = 7

This represents the horizontal asymptote (it has no vertical asymptote)

Hence, the horizontal asymptote is y = 7 and the function is an example of exponential decay

Determine the domain and range of the function.

The function can take any input

So, the domain is -∝ < x < ∝

We have the horizontal asymptote to be

y = 7

The function cannot equal or exceed this value.

So, the range is y < 7

Calculate the x-intercept and y-intercept, then sketch the function.

Set x = 0

g(0) = -7/8 * 2^(-5 * 0) + 7

This gives

g(0) = -7/8 * 2^(0) + 7

Evaluate the exponent

g(0) = -7/8 + 7

Evaluate the sum

g(0) = 49/8

So, the y-intercept is 49/8

Set g(x) = 0

0 = -7/8 * 2^(-5x) + 7

This gives

-7 = -7/8 * 2^(-5x)

Divide by -7

1 = 1/8 * 2^(-5x)

Multiply by 8

8 = 2^(-5x)

Solve for x

x = -0.6

So, the x-intercept is -0.6

See attachment for the sketch

Read more about exponential functions at:

brainly.com/question/2456547

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