Problem 29 - College Algebra

Question

For the following exercises, find the average rate of change of each function on the interval specified. $$h(x)=5-2 x^{2} \text { on }[-2,4]$$

Step-by-Step Solution

Answer

The average rate of change is -4.

Step 1: Understand the Average Rate of Change Formula

The average rate of change of a function between two points \(a\) and \(b\) is given by the formula \(\frac{f(b) - f(a)}{b - a}\). In this case, we want to find the average rate of change of the function \(h(x) = 5 - 2x^2\) on the interval \([-2, 4]\).

Step 2: Evaluate the Function at the Endpoints

Calculate \(h(-2)\) and \(h(4)\) using the function \(h(x) = 5 - 2x^2\).\[ h(-2) = 5 - 2(-2)^2 = 5 - 2 \cdot 4 = 5 - 8 = -3 \]\[ h(4) = 5 - 2(4)^2 = 5 - 2 \cdot 16 = 5 - 32 = -27 \]

Step 3: Substitute Values into the Average Rate of Change Formula

Using the values calculated in Step 2, substitute them into the average rate of change formula:\[ \text{Average rate of change} = \frac{h(4) - h(-2)}{4 - (-2)} = \frac{-27 - (-3)}{4 + 2} = \frac{-27 + 3}{6} = \frac{-24}{6} = -4 \]

Step 4: Interpret the Result

The average rate of change of the function \(h(x) = 5 - 2x^2\) from \(-2\) to \(4\) is \(-4\). This means that for each unit increase in \(x\), the function \(h(x)\) decreases by 4 units on average over the interval.

Key Concepts

Quadratic Functions

A quadratic function is a type of polynomial function that can be represented in the standard form as \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). These functions form a parabola when graphed. The shape of the parabola opens upwards when \( a > 0 \) and opens downward if \( a < 0 \). For our function, \( h(x) = 5 - 2x^2 \), the coefficient of \( x^2 \) is \(-2\), indicating a downward-opening parabola.
This downward direction suggests that as \( x \) increases or decreases, the \( y \) value will initially increase to a maximum point and then fall. Understanding these characteristics is crucial for solving various problems, including finding the average rate of change over an interval.

Interval Notation

Interval notation is a method of representing a set of numbers between two endpoints. It is a compact way to express ranges of numbers without listing them. It uses brackets to define whether endpoints are included or not:

• ")" and "(" indicate that the endpoint is not included (open interval).
• "]" and "[" indicate that the endpoint is included (closed interval).

For example, the interval \([-2, 4]\) includes both -2 and 4, as shown by the square brackets. This notation lets us easily interpret where function evaluations will occur in problems related to averages or changes over a specified range. Understanding how to read and write interval notation is crucial for rephrasing and interpreting mathematical statements across many areas of math.

Function Evaluation at Points

Function evaluation involves calculating the output of a function based on specific input values. For any given function \( f(x) \), evaluating it at a point means substituting the input \( x \) into the expression to find \( f(x) \). This is necessary for tasks such as calculating the average rate of change.
In the exercise, the task required us to evaluate the function \( h(x) = 5 - 2x^2 \) at specific points \(-2\) and \(4\). With \( h(-2) = -3 \) and \( h(4) = -27 \), these outputs were crucial for applying the average rate of change formula. Knowing how to evaluate functions at specific points is vital for understanding function behavior and deriving various mathematical properties.