Answer:
[tex]k = -7[/tex]
Explanation:
Given
[tex]f(x) = 5^x[/tex] through (0,1) and (1,5)
[tex]g(x) = 5^x + k[/tex] 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 [tex]g(x) = 5^x + k[/tex]
[tex]g(x) = 5^x + k[/tex]
[tex]-6 = 5^0 + k[/tex]
[tex]-6 = 1 + k[/tex]
[tex]k + 1 = -6[/tex]
[tex]k = -6 - 1[/tex]
[tex]k = -7[/tex]
For (1,-2) --- Substitute 1 for x and -2 for g(x) in [tex]g(x) = 5^x + k[/tex]
[tex]g(x) = 5^x + k[/tex]
[tex]-2 = 5^1 + k[/tex]
[tex]-2 = 5 + k[/tex]
[tex]k+5 = -2[/tex]
[tex]k= -2-5[/tex]
[tex]k = -7[/tex]
To further show that k = -7, we have the following:
[tex]f(x) = 5^x[/tex]
When translated downward by 7 units, the function becomes
[tex]g(x) = f(x) - 7[/tex]
Recall that: [tex]g(x) = 5^x + k[/tex] and [tex]f(x) = 5^x[/tex]
So, we have:
[tex]5^x + k = 5^x - 7[/tex]
Subtract [tex]5^x[/tex] from both sides
[tex]5^x - 5^x + k = 5^x - 5^x + 7[/tex]
[tex]k = -7[/tex]
