Question

On a coordinate plane, 2 exponential functions are shown. f (x) = 5 Superscript x Baseline approaches y = 0 in quadrant 2 and increases in quadrant 1. It goes though (0, 1) and (1, 5). g (x) = 5 Superscript x Baseline + k approaches y = negative 7.5 in quadrant 3 and increases into quadrant 4 going through (0, negative 6) and (1, negative 2).

The graph of f(x) was vertically translated down by a value of k to get the function g(x) = 5x + k. What is the value of k?

−7
−6
5
7

Answer 1

(A) -7

Step-by-step explanation:

Answer 2

`k = -7`

Step-by-step explanation:

Given

`f(x) = 5^x` through (0,1) and (1,5)

`g(x) = 5^x + k` through (0,-6) and (1,-2)

Required

Determine the value of k if f(x) is translated vertically down

To get the value of k, we perform the following operations.

For (0,-6) --- Substitute 0 for x and -6 for g(x) in
`g(x) = 5^x + k`

`g(x) = 5^x + k`

`-6 = 5^0 + k`

`-6 = 1 + k`

`k + 1 = -6`

`k = -6 - 1`

`k = -7`

For (1,-2) --- Substitute 1 for x and -2 for g(x) in
`g(x) = 5^x + k`

`g(x) = 5^x + k`

`-2 = 5^1 + k`

`-2 = 5 + k`

`k+5 = -2`

`k= -2-5`

`k = -7`

To further show that k = -7, we have the following:

`f(x) = 5^x`

When translated downward by 7 units, the function becomes

`g(x) = f(x) - 7`

Recall that:
`g(x) = 5^x + k` and
`f(x) = 5^x`

So, we have:

`5^x + k = 5^x - 7`

Subtract
`5^x` from both sides

`5^x - 5^x + k = 5^x - 5^x + 7`

`k = -7`