Find the equation of the exponential function represented by the table below:

x*x*y*y*

00

55

11

2, point, 52.5

22

1, point, 2, 51.25

33

0, point, 6, 2, 50.625

To find the equation of an exponential function from a given table of values, we can start with the general form of an exponential function:

\[
y = ab^x
\]

Where:
- \( a \) is the initial value (when \( x = 0 \)),
- \( b \) is the base or growth factor,
- \( x \) is the input variable,
- \( y \) is the output.

### Step 1: Extract Information from the Table
Here’s what the table suggests:

| \( x \)  | \( y \)    |
|---------|------------|
| 0       | 5          |
| 1       | 2.5        |
| 2       | 1.25       |
| 3       | 0.625      |

### Step 2: Use the First Point
From the table, when \( x = 0 \), \( y = 5 \). This gives us the value of \( a \):

\[
a = 5
\]

So the equation so far is:

\[
y = 5b^x
\]

### Step 3: Solve for \( b \) Using Other Points
Now, we can use the point where \( x = 1 \) and \( y = 2.5 \) to find \( b \).

Substitute \( x = 1 \) and \( y = 2.5 \) into the equation:

\[
2.5 = 5b^1
\]

Solve for \( b \):

\[
b = \frac{2.5}{5} = 0.5
\]

### Step 4: Write the Final Equation
Now that we have both \( a = 5 \) and \( b = 0.5 \), the equation of the exponential function is:

\[
y = 5(0.5)^x
\]

### Final Answer: 
The equation of the exponential function is:

\[
y = 5(0.5)^x
\]

Would you like more details on how this was done or have any questions about the steps? Here's some related content you might find helpful:

1. How to find the general form of an exponential equation.
2. What is the significance of the base \( b \) in exponential growth/decay?
3. How to identify exponential growth or decay from a table.
4. How to solve for \( b \) with other points if \( x = 0 \) is not available.
5. The difference between exponential and linear functions.

**Tip**: If you have an exponential function and you're unsure about the base, always use two points and substitute them into the equation to find \( b \).

Find the equation of the exponential function represented by the table below:

x    y
0    5
1    2.5
2    1.25
3    0.625

Learn how to derive the equation of an exponential function using a table of values. This problem involves using the general form y = ab^x and solving for the parameters 'a' and 'b'. The example provides step-by-step instructions with points (0,5), (1,2.5), (2,1.25), and (3,0.625). Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation