Question

graph $y = -(4)^x$.

Explanation:

Step1: Analyze the function type

The function \( y = - (4)^x \) is an exponential function. The base \( 4>1 \), so the parent function \( y = 4^x \) is an increasing exponential function. When we multiply by - 1, it reflects the graph of \( y = 4^x \) over the x - axis.

Step2: Analyze the behavior at key points

• When \( x = 0 \), \( y=-4^{0}=- 1\). So the y - intercept is \( (0, - 1) \).
• As \( x

ightarrow+\infty \), \( 4^{x}
ightarrow+\infty \), so \( y=-4^{x}
ightarrow-\infty \).

• As \( x

ightarrow-\infty \), \( 4^{x}
ightarrow0 \) (since for exponential functions with base \( a > 1 \), \( a^{x}
ightarrow0 \) as \( x
ightarrow-\infty \)), so \( y=-4^{x}
ightarrow0 \).

Step3: Compare with the given graphs

• The first graph is increasing (since as x increases, y increases), but our function should be decreasing (because of the negative sign) for \( x>0 \) and approaching 0 as \( x

ightarrow-\infty \) and \( -\infty \) as \( x
ightarrow+\infty \), so it is not the correct graph.

• The second graph is a parabola - like shape (it looks like a quadratic or even - function graph), but our function is an exponential function, so it is not the correct graph.
• The third graph: At \( x = 0 \), \( y=-1 \) (matches our y - intercept). As \( x

ightarrow+\infty \), \( y
ightarrow-\infty \) (matches our analysis) and as \( x
ightarrow-\infty \), \( y
ightarrow0 \) (matches our analysis). Also, it is a decreasing exponential - like graph (after reflection), so this graph is the correct one.

Answer:

The third graph (the one at the bottom)