Question

Use the image to answer the question.

A concave-up, increasing curve is plotted on a coordinate plane where the x-axis ranges from negative 2 to 4 in increments of 2 and the y-axis ranges from 0 to 20 in increments of 5. The curve passes through 3 marked points.

Use the graph to fill in the missing elements of the exponential growth equation.

(1 point)
y=
(
)x

Answer 1

To fill in the missing elements for the exponential growth equation \( y = a \cdot b^x \), we need to determine the values of \( a \) and \( b \) based on the graph you've described. Since I cannot see the specific marked points on the graph, I will give you a general way to find \( a \) and \( b \):

1. **Identify a point on the curve (x, y)**: Look for the coordinates of the point where the curve intersects the graph. This can be your first point.

2. **Identify another point on the curve as a second data point (x', y')**: Find at least one more point on the curve to help determine the growth rate.

3. **Determine \( a \)**: The value of \( a \) represents the initial value when \( x = 0 \). If there is a point where \( x = 0 \) on your graph, \( a \) is simply the \( y \)-coordinate of that point.

4. **Calculate \( b \)**: To calculate \( b \) using the points (x, y) and (x', y'), you can rearrange the equation:
\[
\frac{y'}{y} = b^{x' - x}
\]
From this equation, you can solve for \( b \):
\[
b = \left( \frac{y'}{y} \right)^{\frac{1}{x' - x}}
\]

5. **Complete the equation**: Once you have \( a \) and \( b \), substitute these into the formula \( y = a \cdot b^x \).

If you can provide the specific \( x \) and \( y \) values from the points marked on the curve, I can help you determine the specific values for \( a \) and \( b \).