Question

Zena planted a bamboo stalk in her backyard. The height of the plant increased in height over the next few months. Which exponential equation in the form y=abx
represents the situation shown on the graph?
graph shows a line passing through (0,3) (1,4.5) (2,6.75)


(1 point)
Responses

y=3(32)x
y is equal to 3 times 3 halves to the x th power

y=3(23)x
y is equal to 3 times 2 thirds to the x th power

y=4.5(52)x
y is equal to 4 point 5 times 5 halves to the x th power

y=4.5(3)x

Answer 1

To determine the correct exponential equation that represents the growth in the height of Zena's bamboo stalk, we can analyze the points given on the graph: (0, 3), (1, 4.5), and (2, 6.75).

1. The point (0, 3) indicates that when x = 0, y = 3. This suggests that **a = 3** in the equation \(y = ab^x\).

2. To find \(b\), we can use the second point (1, 4.5):
\[
y = 3b^1 = 4.5
\]
Dividing both sides by 3:
\[
b = \frac{4.5}{3} = \frac{3}{2}
\]

3. Now we can check if this \(b\) value works for the third point (2, 6.75):
\[
y = 3b^2 = 6.75
\]
Substituting \(b = \frac{3}{2}\):
\[
y = 3 \left(\frac{3}{2}\right)^2 = 3 \cdot \frac{9}{4} = \frac{27}{4} = 6.75
\]
This confirms that our values are accurate.

Therefore, the equation representing the situation in the exponential form \(y = ab^x\) is:
\[
y = 3 \left(\frac{3}{2}\right)^x
\]
This matches option:
**y is equal to 3 times \( \frac{3}{2} \) to the x th power.**