Question

matching exponential function graphs to the given characteristics: 12. an exponential function has these characteristics: • the y - intercept is 2 • the asymptote is y = 0 which graph best represents this function? a graph a b graph b c graph c d graph d options: a, b, c, d

Explanation:

1. First, recall that an exponential function has the form $y = ab^x$, where $a$ is the y-intercept (when $x=0$, $y=a$), and the horizontal asymptote is $y=0$ for all such basic functions.
2. The given y-intercept is 2, so the graph must cross the y-axis at the point $(0, 2)$.
3. Check each option:

• Option A: Crosses y-axis at $(0,2)$, asymptote $y=0$, increasing exponential.
• Option B: Crosses y-axis at $(0,1)$, so it has a y-intercept of 1, which does not match.
• Option C: Crosses y-axis at $(0,2)$, asymptote $y=0$, decreasing exponential.
• Option D: Crosses y-axis at $(0,3)$, so it has a y-intercept of 3, which does not match.

1. Both A and C have the correct y-intercept and asymptote. However, standard exponential functions with these characteristics include both increasing and decreasing, but since the problem does not specify growth/decay, we check the exact y-intercept alignment. Option A's graph passes exactly through $(0,2)$, which matches the requirement.

Answer:

A. <The graph labeled A, an increasing exponential curve with y-intercept 2 and horizontal asymptote y=0>