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)=

1 answer

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.

Similar Questions
  1. Use the image to answer the question.A curve labeled b of x and five points on the curve are plotted on a coordinate plane. The
    1. answers icon 1 answer
  2. Use the image to answer the question.A curve labeled q of x and five points on the curve are plotted on a coordinate plane. The
    1. answers icon 1 answer
  3. Use the image to answer the question.A curve labeled b of x and five points on the curve are plotted on a coordinate plane. The
    1. answers icon 1 answer
  4. Use the image to answer the question.A curve labeled m of x and seven points on the curve are plotted on a coordinate plane. The
    1. answers icon 1 answer
more similar questions