How do you determine the y intercept of an exponential function f(x)=ab^x+k that has been both stretched and translated

To find the y-intercept of the exponential function [tex]\bold{\sf f(x) = ab^x + k}[/tex] that has been both stretched (or compressed) and translated, we can follow these steps:

Understand the Form of the Exponential Function:

The general form of an exponential function is:

[tex]\bold{\sf f(x) = ab^x + k}[/tex]

In this form:

[tex]\bold{\sf a}[/tex] is the initial value.
[tex]\bold{\sf b}[/tex] is the base, representing the stretching or compression factor.
[tex]\bold{\sf k}[/tex] is the vertical translation.

Evaluate at [tex]\bold{\sf x = 0}[/tex]:

To find the y-intercept, substitute [tex]\bold{\sf x = 0}[/tex] into the function. This is because the y-intercept occurs when [tex]\bold{\sf x = 0}[/tex].

[tex]\sf f(0) = ab^0 + k[/tex]

[tex]\sf f(0) = a + k[/tex]

Interpret the Result:

The value obtained [tex]\bold{\sf (a + k)}[/tex] gives us the y-intercept after both stretching or compressing and translating.

The term [tex]\bold{\sf a}[/tex] represents the initial value, and [tex]\bold{\sf k}[/tex] represents the vertical translation.

Therefore, the y-intercept of an exponential function [tex]\sf f(x)=ab^x+k [/tex] that has been both stretched and translated is:

[tex] \fbox{\sf (0,(a+k))}[/tex]