Question

identify the transformations made on f(x) = e^x to create the graph of g(x) = -\frac{1}{2}e^{(x + 6)} + 4\
options:\

• vertical compression by a factor of 1/2, reflection over y - axis, left 6, down 4.\
• vertical compression by a factor of 1/2, reflection over x - axis, right 6, up 4.\
• vertical stretch by a factor of 2, reflection over x - axis, left 6, up 4.\
• vertical compression by a factor of 1/2, reflection over x - axis, left 6, up 4.

Explanation:

Step1: Analyze vertical transformation

The function \( f(x) = e^x \) becomes \( g(x)=-\frac{1}{2}e^{(x + 6)}+4 \). The coefficient of \( e^{(x+6)} \) is \( -\frac{1}{2} \). The negative sign indicates a reflection over the x - axis, and the factor \( \frac{1}{2} \) (since \( 0<\frac{1}{2}<1 \)) indicates a vertical compression by a factor of \( \frac{1}{2} \).

Step2: Analyze horizontal transformation

For the exponent, we have \( (x + 6) \) which is of the form \( (x - h) \) with \( h=-6 \). In function transformations, \( f(x - h) \) shifts the graph of \( f(x) \) left by \( |h| \) units when \( h<0 \). So \( e^{(x+6)}=e^{x-(-6)} \), which means a shift left by 6 units.

Step3: Analyze vertical shift

The \( + 4 \) at the end of the function \( g(x) \) indicates a vertical shift up by 4 units.

Answer:

The correct option is the one with "vertical compression by a factor of 1/2, reflection over x - axis, left 6, up 4" (the teal - colored option in the original problem).