Question

graph the exponential function
( y = -3^x )

identify the graph of the function ( y = -3^x )

\\( \bigcirc \\) a.
graph a
\\( \bigcirc \\) b.
graph b
\\( \bigcirc \\) c.
graph c
\\( \bigcirc \\) d.
graph d

Explanation:

Step1: Analyze the parent function

The parent function of \( y = -3^x \) is \( y = 3^x \), which is an exponential growth function (since the base \( 3>1 \)) with a y - intercept at \( (0,1) \) (when \( x = 0 \), \( y=3^0 = 1 \)) and it increases as \( x \) increases.

Step2: Analyze the transformation

For the function \( y=-3^x \), we have a reflection over the x - axis of the parent function \( y = 3^x \). When \( x = 0 \), \( y=-3^0=- 1 \), so the y - intercept is \( (0, - 1) \). As \( x\to+\infty \), \( 3^x\to+\infty \), so \( y=-3^x\to-\infty \). As \( x\to-\infty \), \( 3^x\to0 \) (because for \( a>1 \), \( a^x\to0 \) as \( x\to-\infty \)), so \( y =-3^x\to0 \).

Now let's analyze the options:

• Option A: The graph seems to approach 0 as \( x\to+\infty \) and has a y - intercept above 0, which does not match \( y=-3^x \) (y - intercept is - 1).
• Option B: The graph has a y - intercept at \( (0, - 1) \) (or near - 1), as \( x\) increases, the function goes to \( -\infty \), and as \( x\) decreases, the function approaches 0. This matches the behavior of \( y=-3^x \).
• Option C: This graph is an exponential growth (increasing as \( x\) increases) with y - intercept near 0, which is the parent function \( y = 3^x \) (or similar growth), not the reflected one.
• Option D: This graph is also an exponential growth (increasing as \( x\) increases) with y - intercept above 0, not matching \( y=-3^x \).

Answer:

B