/*! 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 81 Use the fact that a quadratic fu... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 81

Use the fact that a quadratic function of the form \(f(x)=a x^{2}+b x+c\) with \(b^{2}-4 a c>0\) may also be written in the form \(f(x)=a\left(x-r_{1}\right)\left(x-r_{2}\right),\) where \(r_{1}\) and \(r_{2}\) are the \(x\) -intercepts of the graph of the quadratic function. (a) Find quadratic functions whose \(x\) -intercepts are -3 and 1 with \(a=1 ; a=2 ; a=-2 ; a=5\) (b) How does the value of \(a\) affect the intercepts? (c) How does the value of \(a\) affect the axis of symmetry? (d) How does the value of \(a\) affect the vertex? (e) Compare the \(x\) -coordinate of the vertex with the midpoint of the \(x\) -intercepts. What might you conclude?

Short Answer

Expert verified
The vertex's x-coordinate is always the midpoint of the x-intercepts. The value of 'a' affects the y-coordinate of the vertex.

Step by step solution

01

Express the function using intercepts

Start by using the given quadratic intercept form, where the quadratic function is written as \[ f(x) = a(x - r_1)(x - r_2) \]. Given \( r_1 = -3 \) and \( r_2 = 1 \), we substitute to get \[ f(x) = a(x + 3)(x - 1) \] for different values of \( a \).
02

Find quadratic functions

Now compute the quadratic functions for each value of \( a \):For \( a = 1 \): \[ f(x) = (x + 3)(x - 1) \] expand to get \[ f(x) = x^2 + 2x - 3 \].For \( a = 2 \): \[ f(x) = 2(x + 3)(x - 1) \] expand to get \[ f(x) = 2x^2 + 4x - 6 \].For \( a = -2 \): \[ f(x) = -2(x + 3)(x - 1) \] expand to get \[ f(x) = -2x^2 - 4x + 6 \].For \( a = 5 \): \[ f(x) = 5(x + 3)(x - 1) \] expand to get \[ f(x) = 5x^2 + 10x - 15 \].
03

Effect of 'a' on intercepts

The value of \( a \) does not affect the intercepts \( r_1 \) and \( r_2 \), which remain \( -3 \) and \( 1 \) respectively, as the intercepts are solely determined by the roots of the equation.
04

Effect of 'a' on axis of symmetry

The axis of symmetry of a quadratic function \( f(x) = a(x - r_1)(x - r_2) \) is given by \[ x = \frac{r_1 + r_2}{2} \].For \( r_1 = -3 \) and \( r_2 = 1 \), the axis of symmetry is \[ x = \frac{-3 + 1}{2} = \frac{-2}{2} = -1 \].Thus, the value of \( a \) does not affect the axis of symmetry.
05

Effect of 'a' on the vertex

The vertex of the quadratic function \( f(x) = a(x - r_1)(x - r_2) \) lies on the axis of symmetry at \[ x = \frac{r_1 + r_2}{2} = -1 \].Plugging \( x = -1 \) into the quadratic function for different values of \( a \) gives:For \( a = 1 \): \[ f(x) = (x + 3)(x - 1) \]: \[ f(-1) = 1(( -1 + 3)(-1 - 1)) = -4 \].For \( a = 2 \): \[ f(x) = 2(x + 3)(x - 1) \]: \[ f(-1) = 2(( -1 + 3)(-1 - 1)) = -8 \].For \( a = -2 \): \[ f(x) = -2(x + 3)(x - 1) \]: \[ f(-1) = -2(( -1 + 3)(-1 - 1)) = 8 \].For \( a = 5 \): \[ f(x) = 5(x + 3)(x - 1) \]: \[ f(-1) = 5(( -1 + 3)(-1 - 1)) = -20 \].The value of \( a \) affects the \( y \)-coordinate of the vertex but not the \( x \)-coordinate.
06

Comparison of x-coordinate of the vertex with midpoint of intercepts

The \( x \)-coordinate of the vertex is calculated as \[ \frac{r_1 + r_2}{2} \],which equals \[ -1 \].Thus, the \( x \)-coordinate of the vertex is indeed the midpoint of the \( x \)-intercepts, leading to the conclusion that the vertex lies exactly midway between the x-intercepts.

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.

x-intercepts
The x-intercepts of a quadratic function are the points where the graph intersects the x-axis. These intercepts are found by solving the equation f(x) = 0. For a quadratic function of the form f(x) = a(x - r1)(x - r2), the x-intercepts are r1 and r2.

In the given example, the intercepts are -3 and 1. This means the graph of the function touches the x-axis at x = -3 and x = 1. Here are the quadratic functions for different values of 'a':
  • For a = 1: f(x) = (x + 3)(x - 1) = x^2 + 2x - 3
  • For a = 2: f(x) = 2(x + 3)(x - 1) = 2x^2 + 4x - 6
  • For a = -2: f(x) = -2(x + 3)(x - 1) = -2x^2 - 4x + 6
  • For a = 5: f(x) = 5(x + 3)(x - 1) = 5x^2 + 10x - 15
Notice that changing 'a' only affects the stretching or compressing of the parabola but not the x-intercepts themselves.
axis of symmetry
The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the parabola and divides it into two mirror-image halves. This line can be found using the formula x = (r1 + r2)/2, where r1 and r2 are the x-intercepts.

In the example provided, with x-intercepts at -3 and 1, we calculate the axis of symmetry as follows:
  • Axis of symmetry x = (-3 + 1)/2 = -1
This means that no matter what value 'a' takes, the axis of symmetry for this function will always be x = -1.
vertex
The vertex of a quadratic function is the highest or lowest point on its graph, a parabola. For a function in the form f(x) = a(x - r1)(x - r2), the vertex lies on the axis of symmetry x = (r1 + r2)/2.

In our example with intercepts -3 and 1, the axis of symmetry is x = -1. Substituting x = -1 into the quadratic functions gives us the y-coordinate of the vertex for different 'a' values:
  • For a = 1: f(-1) = 1(-1 + 3)(-1 - 1) = -4
  • For a = 2: f(-1) = 2(-1 + 3)(-1 - 1) = -8
  • For a = -2: f(-1) = -2(-1 + 3)(-1 - 1) = 8
  • For a = 5: f(-1) = 5(-1 + 3)(-1 - 1) = -20
Notice that the x-coordinate of the vertex remains constant at -1, but the y-coordinate changes based on the value of 'a'. This implies that 'a' affects the vertical stretching or compressing of the parabola, altering the y-value of the vertex while keeping the x-value the same.

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

Temperature Conversion The linear function \(F(C)=\frac{9}{5} C+32\) converts degrees Celsius to degrees Fahrenheit, and the linear function \(R(F)=F+459.67\) converts degrees Fahrenheit to degrees Rankine. Find a linear function that converts degrees Rankine to degrees Celsius.

Use the fact that a quadratic function of the form \(f(x)=a x^{2}+b x+c\) with \(b^{2}-4 a c>0\) may also be written in the form \(f(x)=a\left(x-r_{1}\right)\left(x-r_{2}\right),\) where \(r_{1}\) and \(r_{2}\) are the \(x\) -intercepts of the graph of the quadratic function. (a) Find quadratic functions whose \(x\) -intercepts are -5 and 3 with \(a=1 ; a=2 ; a=-2 ; a=5\) (b) How does the value of \(a\) affect the intercepts? (c) How does the value of \(a\) affect the axis of symmetry? (d) How does the value of \(a\) affect the vertex? (e) Compare the \(x\) -coordinate of the vertex with the midpoint of the \(x\) -intercepts. What might you conclude?

The following data represent the various combinations of soda and hot dogs that Yolanda can buy at a baseball game with \(\$60$$\begin{array}{|cc|}\hline \text { Soda, } s & \text { Hot Dogs, } h \\\\\hline 20 & 0 \\\15 & 3 \\\10 & 6 \\\5 & 9 \\\\\hline\end{array}$$ (a) Plot the ordered pairs \)(s, h)\( in a Cartesian plane. (b) Show that the number \)h\( of hot dogs purchased is a linear function of the number \)s\( of sodas purchased. (c) Determine the linear function that describes the relation between \)s\( and \)h$ (d) What is the domain of the linear function? (e) Graph the linear function in the Cartesian plane drawn in part (a). (f) Interpret the slope. (g) Interpret the intercepts.

True or False The correlation coefficient is a measure of the strength of a linear relation between two variables and must lie between -1 and 1 , inclusive.

The daily revenue \(R\) achieved by selling \(x\) boxes of candy is \(R(x)=9.5 x-0.04 x^{2}\). The daily cost \(C\) of selling \(x\) boxes of candy is \(C(x)=1.25 x+250 .\) (a) How many boxes of candy must the firm sell to maximize revenue? What is the maximum revenue? (b) Profit is given as \(P(x)=R(x)-C(x) .\) What is the profit function? (c) How many boxes of candy must the firm sell to maximize profit? What is the maximum profit? (d) Provide a reasonable explanation as to why the answers found in parts (a) and (c) differ. Explain why a quadratic function is a reasonable model for revenue.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

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.