AP Precalculus Practice Test: Unit 1 Question #8 Rate of Change from a Table

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**AP Precalculus Practice Test: Unit 1, Question #8: Rate of Change from a Table**

In this question, you are asked to find the rate of change of a function from a table of values. The rate of change can be calculated by determining how much the function's output (dependent variable) changes as the input (independent variable) changes.

### 1. **Understanding Rate of Change**:
The **rate of change** between two points is essentially the change in the output value divided by the change in the input value. It is calculated using the formula:

\[
\text{Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

Where:
- \( f(x_1) \) and \( f(x_2) \) are the function values (outputs) at points \( x_1 \) and \( x_2 \),
- \( x_1 \) and \( x_2 \) are the input values (independent variables).

### 2. **Steps to Calculate Rate of Change from a Table**:

#### Step 1: Identify the Two Points
From the table, choose two points \( (x_1, f(x_1)) \) and \( (x_2, f(x_2)) \) for which you want to calculate the rate of change. These points should have different values for \( x \) so that you can calculate the change in both the \( x \)-values and the \( f(x) \)-values.

#### Step 2: Apply the Rate of Change Formula
Substitute the values from the table into the formula:

\[
\text{Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

#### Step 3: Simplify the Result
Simplify the expression to find the rate of change.

### 3. **Example Problem**:

Suppose the table of values for a function \( f(x) \) is as follows:

| \( x \) | \( f(x) \) |
|--------|------------|
| 1 | 3 |
| 3 | 7 |
| 5 | 11 |
| 7 | 15 |

You are asked to find the rate of change between \( x = 1 \) and \( x = 5 \).

#### Step 1: Identify the Two Points
We select the points where \( x = 1 \) and \( x = 5 \):
- \( (x_1, f(x_1)) = (1, 3) \)
- \( (x_2, f(x_2)) = (5, 11) \)

#### Step 2: Apply the Rate of Change Formula
Now, substitute these values into the formula:

\[
\text{Rate of Change} = \frac{f(5) - f(1)}{5 - 1} = \frac{11 - 3}{5 - 1} = \frac{8}{4} = 2
\]

#### Step 3: Interpret the Result
The rate of change between \( x = 1 \) and \( x = 5 \) is \( 2 \). This means that for every increase of 1 unit in \( x \), the function's output \( f(x) \) increases by 2 units.

### 4. **Conclusion**:
To calculate the rate of change from a table:
1. Choose two points from the table with different \( x \)-values.
2. Use the rate of change formula \( \frac{f(x_2) - f(x_1)}{x_2 - x_1} \).
3. Simplify the result to find the rate of change between the two points. This tells you how much the output changes for each unit change in the input.

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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