/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 77 Find the average rate of change ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 77

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\). $$ \text {Function} \quad \text {x-Values}$$ $$ f(x)=x^{2}+12 x-4 \quad x_{1}=1, x_{2}=5 $$

Short Answer

Expert verified
The average rate of change of the function from \(x_{1}\) to \(x_{2}\) is 13.

Step by step solution

01

Substitution of \(x_{1}\) into \(f(x)\)

First, let's find the function values for \(x_{1}\) which is 1. Plug \(x_{1} = 1\) into the function, \(f(x) = x^{2}+12x-4\), yielding: \(f(x_{1}) = (1)^{2} + 12*1 - 4 = 9\)
02

Substitution of \(x_{2}\) into \(f(x)\)

Next, let's find the function values for \(x_{2}\) which is 5. Plug \(x_{2} = 5\) into the function, \(f(x) = x^{2}+12x-4\), yielding: \(f(x_{2}) = (5)^{2} + 12*5 - 4 = 61\)
03

Calculation of the Average Rate of Change

Now that we have both \(f(x_{1})\) and \(f(x_{2})\), let's calculate the average rate of change using the formula \((f(x_{2}) - f(x_{1})) / (x_{2} - x_{1})\). This results to: \((61 - 9) / (5 - 1) = 52/4 = 13\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Evaluation
To evaluate a function, we must substitute given values into the function equation to determine their specific outputs. Imagine a function as a machine that processes inputs to produce outputs. If we know the input, in our case specific x-values, we can find the corresponding output or "function value."
  • Identify the given x-values: In our exercise, these are \(x_1 = 1\) and \(x_2 = 5\).
  • Input each x-value into the function \(f(x) = x^2 + 12x - 4\).
  • Calculate the output values, which are \(f(1)\) and \(f(5)\).
Function evaluation helps determine values necessary for further procedures like finding rates of change.
Substitution
Substitution is a straightforward yet crucial mathematical technique used when evaluating functions. It involves replacing variables with specific values or expressions to simplify the function and facilitate calculations.
In the context of evaluating a quadratic function:
  • Write the function: \(f(x) = x^2 + 12x - 4\).
  • Substitute the x-value, such as \(x=1\) or \(x=5\).
  • Replace every instance of the variable x with the given number in the equation.
  • Calculate the result of the newly simplified expression.
Substitution tells us exactly what a specific part of the function looks like when x reaches a particular value, providing crucial insights into the function's behavior.
Quadratic Functions
Quadratic functions form an essential class of polynomial functions, defined by their standard form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a \/neq 0\).
These functions graph as parabolas, exhibiting various characteristics:
  • Vertex: The highest or lowest point on the graph.
  • Axis of Symmetry: A line that divides the parabola into two mirror images, running vertically through the vertex.
  • Opening direction: Upward if \(a > 0\) and downward if \(a < 0\).
Analyzing a quadratic function like \(f(x) = x^2 + 12x - 4\) reveals its shape and behavior, crucial for calculating rates like changes over specific intervals.
Rate of Change Formula
The rate of change formula is used to gauge how a function value changes as the x-value changes over an interval. This is the essence of calculating average rates of change.
The formula used is:\[\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}\]
  • Determine \(f(x_1)\) and \(f(x_2)\) by evaluating the function at the given x-values.
  • Subtract these function values to find the change in output.
  • Divide this difference by the change in x-values \((x_2 - x_1)\).
This formula provides insight into the rate at which the function's value increases or decreases over a specific interval. It is especially useful in understanding behavior over segments of quadratic functions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The linear model with the least sum of square differences is called the _____ _____ _____ line.

Determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=|x-2|, \quad x \leq 2 $$

Assume that \(y\) is directly proportional to \(x\). Use the given \(x\) -value and \(y\) -value to find a linear model that relates \(y\) and \(x\). $$ x=10, y=2050 $$

Find a mathematical model for the verbal statement. \(y\) varies inversely as the square of \(x\)

Use the functions given by \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$ \left(g^{-1} \circ f^{-1}\right)(-3) $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.