Question

Find the average rate of change of \(f(x)=-2 x^{2}+4\) (a) From 0 to 2 (b) From 1 to 3 (c) From 1 to 4

Short answer

(a) -4, (b) -8, (c) -10

Step-by-step solution

Understand the formula for average rate of change

The average rate of change of a function between two points \(x_1\) and \(x_2\) is given by \[ \frac{f(x_2) - f(x_1)}{x_2 - x_1} \].

Calculate for part (a)

First, find \(f(0)\) and \(f(2)\): \[ f(0) = -2(0)^2 + 4 = 4 \] \[ f(2) = -2(2)^2 + 4 = -8 + 4 = -4 \] Now, use the average rate of change formula: \[ \frac{f(2) - f(0)}{2 - 0} = \frac{-4 - 4}{2} = \frac{-8}{2} = -4 \].

Calculate for part (b)

First, find \(f(1)\) and \(f(3)\): \[ f(1) = -2(1)^2 + 4 = -2 + 4 = 2 \] \[ f(3) = -2(3)^2 + 4 = -18 + 4 = -14 \] Now, use the average rate of change formula: \[ \frac{f(3) - f(1)}{3 - 1} = \frac{-14 - 2}{3 - 1} = \frac{-16}{2} = -8 \].

Calculate for part (c)

First, find \(f(1)\) and \(f(4)\): \[ f(1) = -2(1)^2 + 4 = -2 + 4 = 2 \] \[ f(4) = -2(4)^2 + 4 = -32 + 4 = -28 \] Now, use the average rate of change formula: \[ \frac{f(4) - f(1)}{4 - 1} = \frac{-28 - 2}{4 - 1} = \frac{-30}{3} = -10 \].

Key concepts

Quadratic Functions

Quadratic functions are an important concept in algebra and are usually represented in the form \(f(x) = ax^2 + bx + c\). In this case, we are working with a specific quadratic function: \(f(x) = -2x^2 + 4\). This function has certain characteristics:

• The coefficient \(a\) is negative, making the parabola open downwards.
• The function has a constant term of 4, which shifts the parabola upwards.

Understanding the shape of this quadratic function helps in calculating the average rate of change, because it gives you insight into how steeply the function increases or decreases between two points.

Formula Application

To find the average rate of change of a function between two points, we use a specific formula: \(\frac{f(x_2) - f(x_1)}{x_2 - x_1}\). This formula resembles the slope of a line, but instead of just straight lines, it can be applied to any functions, including quadratics.

• Top part (numerator): This represents the change in the function values \(f(x_2) - f(x_1)\).
• Bottom part (denominator): This represents the change in \(x\) values \(x_2 - x_1\).

By applying this formula to each part of our problem, we can understand how \(f(x) = -2x^2 + 4\) changes between different intervals.

Step-by-step Solution

Let's break down the calculations step by step to better understand the average rate of change for each interval.

• Part (a): From 0 to 2
  
  1. Calculate \(f(0)\): \(f(0) = -2(0)^2 + 4 = 4\)
  2. Calculate \(f(2)\): \(f(2) = -2(2)^2 + 4 = -8 + 4 = -4\)
  3. Apply formula: \( \frac{f(2) - f(0)}{2 - 0} = \frac{-4 - 4}{2} = \frac{-8}{2} = -4 \)
  
  Average rate of change from 0 to 2 is -4


• Part (b): From 1 to 3
  
  • Calculate \(f(1)\): \(f(1) = -2(1)^2 + 4 = -2 + 4 = 2\)
  • Calculate \(f(3)\): \(f(3) = -2(3)^2 + 4 = -18 + 4 = -14\)
  • Apply formula: \( \frac{f(3) - f(1)}{3 - 1} = \frac{-14 - 2}{3 - 1} = \frac{-16}{2} = -8 \)
  
  Average rate of change from 1 to 3 is -8


• Part (c): From 1 to 4
  
  • Calculate \(f(1)\): \(f(1) = -2(1)^2 + 4 = -2 + 4 = 2\)
  • Calculate \(f(4)\): \(f(4) = -2(4)^2 + 4 = -32 + 4 = -28\)
  • Apply formula: \( \frac{f(4) - f(1)}{4 - 1} = \frac{-28 - 2}{4 - 1} = \frac{-30}{3} = -10 \)
  
  Average rate of change from 1 to 4 is -10

By carefully following these steps, you can observe how the average rate of change is derived for different segments of the quadratic function \(f(x) = -2x^2 + 4\).