Question

Which graph represents the function f(x) =three-halves(2)x? 1. On a coordinate plane, an exponential function has a horizontal asymptote of y = 0. The function crosses the y-axis at (0, 1) and goes through (1, 3), (2, 9). 2. On a coordinate plane, an exponential function has a horizontal asymptote of y = 0. The function crosses the y-axis at (0, 2) and goes through (1, 3), (2, 4.5), (3, 7) 3. On a coordinate plane, an exponential function has a horizontal asymptote of y = 0. The function crosses the y-axis at (0, 1) and goes through (1, 2), (2, 4), (3, 8). 4. On a coordinate plane, an exponential function has a horizontal asymptote of y = 0. The function crosses the y-axis at (0, 1.5) and goes through (1, 3), (2, 6).

Answer 1

To determine which graph represents the function \( f(x) = \frac{3}{2}(2^x) \), we can analyze the function step by step.

1. **Identify the Y-intercept**:
- The y-intercept is found by evaluating \( f(0) \):
\[
f(0) = \frac{3}{2}(2^0) = \frac{3}{2}(1) = \frac{3}{2} = 1.5.
\]
The function crosses the y-axis at \( (0, 1.5) \).

2. **Evaluate the function at other points**:
- For \( x = 1 \):
\[
f(1) = \frac{3}{2}(2^1) = \frac{3}{2}(2) = 3.
\]
- For \( x = 2 \):
\[
f(2) = \frac{3}{2}(2^2) = \frac{3}{2}(4) = 6.
\]
- For \( x = 3 \):
\[
f(3) = \frac{3}{2}(2^3) = \frac{3}{2}(8) = 12.
\]

3. **Summary of critical points**:
- \( f(0) = 1.5 \) → crosses the y-axis at (0, 1.5)
- \( f(1) = 3 \) → goes through (1, 3)
- \( f(2) = 6 \) → goes through (2, 6)

Now, let's compare these points with the options provided:

1. **Graph 1**: crosses (0, 1), (1, 3), (2, 9) → incorrect.
2. **Graph 2**: crosses (0, 2), (1, 3), (2, 4.5) → incorrect.
3. **Graph 3**: crosses (0, 1), (1, 2), (2, 4) → incorrect.
4. **Graph 4**: crosses (0, 1.5), (1, 3), (2, 6) → **correct**.

Thus, the graph that represents the function \( f(x) = \frac{3}{2}(2^x) \) is **Graph 4**.