Problem 2 - Algebra Form and Function

Question

Find the average rate of change of \(f(x)=x^{2}+3 x\) on the intervals indicated. Between -2 and 4 .

Step-by-Step Solution

Answer

Answer: The average rate of change is 5.

Step 1: Write down the function and interval

The given function is \(f(x) = x^2 + 3x\) and the interval of interest is between -2 and 4.

Step 2: Find the function values at the endpoints

To find the function values at the endpoints, replace 'x' with the endpoint values of the interval in the function. \(f(-2) = (-2)^2 + 3(-2) = 4 - 6 = -2\) \(f(4) = (4)^2 + 3(4) = 16 + 12 = 28\)

Step 3: Find the width of the interval

To find the width of the interval, subtract the lower endpoint value from the higher endpoint value. Width = \(4 - (-2) = 6\)

Step 4: Calculate the average rate of change

To find the average rate of change, divide the difference in function values by the width of the interval. Average rate of change = \(\frac{f(4) - f(-2)}{4 - (-2)} = \frac{28 - (-2)}{6} = \frac{30}{6} = 5\) The average rate of change of the function \(f(x) = x^2 + 3x\) on the interval between -2 and 4 is 5.

Key Concepts

Quadratic Function

A quadratic function is a polynomial function of degree two. Its general form is:

• The standard form: \( f(x) = ax^2 + bx + c \)

Here, \(a\), \(b\), and \(c\) are constants, with \(a \, eq \, 0\). Quadratic functions typically yield a parabola when graphed.

The direction of the parabola (opening upwards or downwards) depends on the sign of \(a\).

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), it opens downwards.

The vertex of the parabola represents the maximum or minimum point of the graph. Understanding a quadratic function's behavior helps in analyzing how the function changes over specific intervals.

Function Evaluation

Function evaluation is a crucial skill in algebra that involves substituting specific values into a function to find the corresponding output. Consider the function \( f(x) = x^2 + 3x \). To evaluate this function at a certain value of \(x\), replace \(x\) with that specific value.

For instance, let's evaluate \( f(x) \) at \(x = -2\):

• Substitute \(-2\) for \(x\) in \( f(x) = x^2 + 3x \).
• Calculate: \( f(-2) = (-2)^2 + 3(-2) = 4 - 6 = -2 \).

This process helps us determine the values of the function at the endpoints, which is essential in problems involving the average rate of change. Evaluating a function accurately will help you understand the behavior and transformation of the function across a specific domain.

Intervals

An interval is a range of values between two specific points and can be either open or closed. For example:

• Open Interval: \((a, b)\) includes all points greater than \(a\) and less than \(b\), but not the endpoints themselves.
• Closed Interval: \([a, b]\) includes all points from \(a\) to \(b\) including the endpoints.

In the context of a function, intervals allow you to study the behavior of the function over specific segments of its domain. It’s particularly important when determining the average rate of change.

In the exercise, we look at the interval from \(-2\) to \(4\) (a closed interval). This means our analysis includes the function values at both \(-2\) and \(4\). Understanding intervals helps in focusing your analysis on particular sections of the function rather than the entire domain, which is crucial for comprehending changes within that section.