Question

identify the transformations made on f(x) = e^x to create the graph of g(x) = -\frac{1}{2}e^{(x + 6)} + 4\
\
vertical compression by a factor of 1/2, reflection over x - axis, left 6, up 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 y - axis, left 6, down 4.

Explanation:

Step1: Analyze vertical transformation

The coefficient of \( e^{(x + 6)} \) is \( -\frac{1}{2} \). The negative sign indicates a reflection over the \( x \)-axis. The absolute value \( \frac{1}{2} \) (since \( 0<\frac{1}{2}<1 \)) indicates a vertical compression by a factor of \( \frac{1}{2} \).

Step2: Analyze horizontal transformation

The exponent is \( (x + 6) \), which is in the form \( (x - h) \) with \( h=-6 \). This means a horizontal shift to the left by 6 units (because \( h=-6 \), shifting left when \( h \) is negative in \( f(x - h) \)).

Step3: Analyze vertical shift

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

Now we match with the options:

• The first option: vertical compression by a factor of \( \frac{1}{2} \), reflection over \( x \)-axis, left 6, up 4. This matches our analysis.
• The second option: right 6 is incorrect (we have left 6).
• The third option: vertical stretch by factor 2 is incorrect (we have compression by \( \frac{1}{2} \)).
• The fourth option: reflection over \( y \)-axis and down 4 are incorrect.

Answer:

The correct option is the first one (the yellow - colored card): vertical compression by a factor of \( \frac{1}{2} \), reflection over \( x \)-axis, left 6, up 4.