Question

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

Explanation:

Step1: Analyze the function type

The function \( y = - (4)^x \) is an exponential function. The parent function is \( y = 4^x \), which is an exponential growth function (since the base \( 4>1 \)) and passes through \( (0,1) \) (when \( x = 0 \), \( y = 4^0=1 \)) and increases as \( x \) increases, approaches \( 0 \) as \( x\to-\infty \).

Step2: Analyze the transformation

The negative sign in front of \( 4^x \) reflects the graph of \( y = 4^x \) over the \( x \)-axis. So for the transformed function \( y=-4^x \):

• When \( x = 0 \), \( y=-4^0=- 1 \), so the graph passes through \( (0, - 1) \).
• As \( x\to+\infty \), \( 4^x\to+\infty \), so \( y=-4^x\to-\infty \).
• As \( x\to-\infty \), \( 4^x\to0 \), so \( y = - 4^x\to0 \).

Step3: Compare with the given graphs

• The first graph is increasing (since as \( x \) increases, \( y \) increases), but our function should be decreasing for \( x>0 \) (since it's a reflection of an increasing exponential), so the first graph is out.
• The second graph is a parabola - like shape (even function, symmetric about \( y \)-axis), but our function is an exponential function, not a polynomial function, so the second graph is out.
• The third graph: when \( x = 0 \), \( y=-1 \) (matches \( y=-4^0=-1 \)), as \( x \) increases (e.g., \( x = 1 \), \( y=-4^1=-4 \); \( x = 2 \), \( y=-4^2=-16 \)), \( y \) decreases (goes more negative), and as \( x\to-\infty \), \( y\to0 \) (the left - hand side of the graph approaches the \( x \)-axis from below). This matches the behavior of \( y=-4^x \).

Answer:

The third graph (the one at the bottom)