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=ab^x
represents the situation shown on the graph?
the main points are (0,3) (1,4.5) (2,6.75)


(1 point)
Responses

y=4.5(5/2)^x
y is equal to 4 point 5 times 5 halves to the x th power

y=3(3/2)^x
y is equal to 3 times 3 halves to the x th power

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

y=4.5(3)^x

Answer 1

To find the correct exponential equation of the form \( y = ab^x \) that fits the given points (0, 3), (1, 4.5), and (2, 6.75), we can analyze the relationships between these points.

1. From the point (0, 3), we know that when \( x = 0 \), \( y = 3 \). Therefore, \( a = 3 \).
2. Now we can write the equation as \( y = 3b^x \).

Next, we can use another point to find \( b \). Let's use the point (1, 4.5):

- From the point (1, 4.5):
\[
y = 3b^1 = 4.5
\]
\[
3b = 4.5
\]
\[
b = \frac{4.5}{3} = 1.5 = \frac{3}{2}
\]

Now we have \( a = 3 \) and \( b = \frac{3}{2} \). Thus, our equation becomes:
\[
y = 3 \left(\frac{3}{2}\right)^x
\]

Now, let's verify with the next point (2, 6.75):

- For \( x = 2 \):
\[
y = 3 \left(\frac{3}{2}\right)^2 = 3 \cdot \frac{9}{4} = \frac{27}{4} = 6.75
\]

This confirms that our equation is correct.

Therefore, the correct exponential equation representing the situation is:
\[
\boxed{y = 3\left(\frac{3}{2}\right)^x}
\]