Question

Use the image to answer the question.

A curve in quadrant 1 of a coordinate plane has 3 plotted points. The x and y axes range from 0 to 10.

Use the information given in the graph to find the exponential function that represents the graph, then find f(5)
. Round the answer to two decimal places.

(2 points)
f(x)=
⋅
x

f(5)=

Answer 1

To find the exponential function that represents the curve given the points in quadrant 1, we generally want to use the form of an exponential function, which is:

\[ f(x) = a \cdot b^x \]

where \( a \) is a constant and \( b \) is the base of the exponential function.

Since I cannot see the image you mentioned, I'll guide you on how to determine the exponential function:

### Steps to Find the Exponential Function

1. **Identify Points**: Look for the x and y coordinates of the plotted points on the graph. Let's assume the points are:
- Point 1: \((x_1, y_1)\)
- Point 2: \((x_2, y_2)\)
- Point 3: \((x_3, y_3)\)

2. **Set Up Equations**: Using the points, set up a system of equations. For each point \((x_i, y_i)\):
- Equation 1: \( y_1 = a \cdot b^{x_1} \)
- Equation 2: \( y_2 = a \cdot b^{x_2} \)
- Equation 3: \( y_3 = a \cdot b^{x_3} \)

3. **Solve for \( a \) and \( b \)**: Choose any two equations and solve for \( a \) and \( b \) simultaneously. This might involve taking logarithms to isolate \( b \).

4. **Confirm with Third Point**: After finding \( a \) and \( b \), plug in the coordinates of the third point to confirm that your function works.

5. **Evaluate \( f(5) \)**: Once you have the function, simply substitute \( x = 5 \) into the function to find \( f(5) \).

### Example

For example, if the points were (1, 3), (2, 6), and (3, 12):

1. Set up the equations:
- \( 3 = a \cdot b^1 \)
- \( 6 = a \cdot b^2 \)
- \( 12 = a \cdot b^3 \)

2. Divide the second equation by the first:
\[
\frac{6}{3} = \frac{a \cdot b^2}{a \cdot b^1} \implies 2 = b
\]

3. Plug \( b \) back into the first equation:
\[
3 = a \cdot 2^1 \implies a = \frac{3}{2}
\]

4. Therefore, our function is:
\[
f(x) = \frac{3}{2} \cdot 2^x
\]

5. Now evaluate \( f(5) \):
\[
f(5) = \frac{3}{2} \cdot 2^5 = \frac{3}{2} \cdot 32 = 48
\]

### Answer

So, if you follow these steps with the actual points you have:

- \( f(x) = a \cdot b^x \)
- \( f(5) = \text{calculated value} \)

Just replace \( a \) and \( b \) with what you find from your points. Don’t forget to round \( f(5) \) to two decimal places as per your requirement.