Problem 48 - Precalculus : Building Concepts and Connections

Question

Find the average rate of change of each ficnetion on the given interval. $$f(x)=2 x^{2}+3 x-1 ; \text { interval: }[-2,-1]$$

Step-by-Step Solution

Answer

The average rate of change of the function \(f(x) = 2x^{2} + 3x - 1\) on the interval [-2,-1] is -3.

Step 1: Find f(-2) and f(-1)

Substitute \(x = -2\) and \(x = -1\) into the function \(f(x)\), in order to get the values of the function at these two points. We have:\(f(-2) = 2(-2)^{2} + 3(-2) - 1 = 2*4 - 6 - 1 = 1\)and \(f(-1) = 2(-1)^{2} + 3(-1) - 1 = 2*1 - 3 - 1 = -2\).

Step 2: Apply the formula for the average rate of change

The average rate of change on the interval [-2,-1] can be found by substituting into the formula \(\frac{f(b) - f(a)}{b - a}\). Here, \(a = -2\) and \(b = -1\). Therefore, we have\(\frac{f(-1) - f(-2)}{-1 - (-2)} = \frac{-2 -1}{-1 - (-2)} = \frac{-3}{1} = -3\).

Key Concepts

Function Evaluation

Function evaluation is a fundamental process in mathematics, integral to figuring out the output of a function for specific inputs. This can be likened to following a recipe, where the function itself is the recipe, the inputs (or independent variables) are the ingredients, and the output (or dependent variable) is the dish created. Let's take a closer look using the quadratic function from our exercise,
In this instance, the function given is \( f(x) = 2x^2 + 3x - 1 \). When asked to evaluate this function at \( x = -2 \) and \( x = -1 \), we're essentially substituting these values in place of \( x \) within the expression:\( f(-2) = 2(-2)^2 + 3(-2) - 1 \) which simplifies to \( f(-2) = 1 \), and for \( f(-1) = 2(-1)^2 + 3(-1) - 1 \) which simplifies to \( f(-1) = -2 \). Through this method, we can find how the function behaves at specific points.

Average Rate of Change Formula

The average rate of change of a function between two points is calculated using a simple formula: \( \frac{f(b) - f(a)}{b - a} \), where \( f(a) \) and \( f(b) \) are the values of the function at points \( a \) and \( b \), respectively. Think of it as measuring the overall slope of the secant line that connects the two points on the graph of the function.
In the context of our exercise, to find the average rate of change for the quadratic function \( f(x) = 2x^2 + 3x - 1 \) on the interval [-2, -1], we've already computed \( f(-2) \) and \( f(-1) \). The next step is to apply the formula using these values and the points \( a = -2 \) and \( b = -1 \):\( \frac{f(-1) - f(-2)}{-1 - (-2)} = \frac{-2 - 1}{1} = -3 \).This tells us that the function decreases by an average of 3 units for every 1 unit increase in \( x \) within the interval from -2 to -1.

Quadratic Function

A quadratic function is an algebraic expression of the second degree, which means it includes a term with the variable raised to the second power (\( x^2 \)). The standard form of a quadratic function is \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. Graphically, quadratic functions are represented by parabolas, which open upwards if \( a > 0 \) and downwards if \( a < 0 \).
The function from our exercise, \( f(x) = 2x^2 + 3x - 1 \), is a classic example of a quadratic function. Due to its second-degree term (\( 2x^2 \)), the graph of this function will form a parabola. The coefficient \( 2 \) indicates the parabola is relatively steep and opens upward since it's positive. Understanding the shape and direction of the graph helps us interpret the average rate of change, as it indicates that the function will have different rates of change as \( x \) changes.