Question

write an equation for f(x) = e^x compressed vertically by a factor of 4/5, shifted right 2 and down 4.
g(x) = \frac{4}{5}e^{x - 2} - 4
g(x) = -5e^{x - 2} - 4
g(x) = \frac{4}{5}e^{x - 2} + 4
g(x) = \frac{4}{5}e^{x + 2} - 4

Explanation:

Step1: Recall transformation rules

For a function \( y = f(x) \), vertical compression by a factor \( a \) (where \( 0 < a < 1 \)) gives \( y = a f(x) \). A horizontal shift right by \( h \) units gives \( y = f(x - h) \), and a vertical shift down by \( k \) units gives \( y = f(x) - k \).

Step2: Apply vertical compression

Given \( f(x)=e^{x} \), vertical compression by a factor of \( \frac{4}{5} \) gives \( y=\frac{4}{5}e^{x} \).

Step3: Apply horizontal shift

Shift right by 2 units: replace \( x \) with \( x - 2 \), so \( y=\frac{4}{5}e^{x - 2} \).

Step4: Apply vertical shift

Shift down by 4 units: subtract 4 from the function, so \( y=\frac{4}{5}e^{x - 2}-4 \).

Answer:

The correct equation is \( g(x)=\frac{4}{5}e^{x - 2}-4 \) (the first option: \( g(x)=\frac{4}{5}e^{x - 2}-4 \))