Precalculus with Limits

3rd Edition

ISBN: 9781133947202

Author: Ron Larson

Publisher: Cengage Learning

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Question

Chapter 2.1, Problem 95E

To determine

To prove: the x -coordinate of the vertex of the graph is the average of the zeros of

$f(x)=ax2+bx+c$ , $a≠0$ .

Expert Solution & Answer

Explanation of Solution

Concept Used:

For the function $f(x)=ax2+bx+c$ with vertex $(−b2a,f(−b2a))$ ,

When $a>0$ , f has a minimum at $x=−b2a$ and the minimum value is $f(−b2a)$ .
When $a<0$ , f has a maximum at $x=−b2a$ and the maximum value is $f(−b2a)$ .

From the above definition it is clear that the x -coordinate of the vertex of the graph is equal to $−b2a$ .

Now to show the x coordinate of the vertex is equal to average of the zeros of $f(x)=ax2+bx+c$

Using quadratic equation to find zeros of the equation $ax2+bx+c=0$ ,

$x=−b±b2−4ac2a$

SO, the two zeros of the equation $ax2+bx+c=0$ are:

$x1=−b+b2−4ac2a$ And $x2=−bb2−4ac2a$

Now find average of these zeros.

$x1+x22=−b+b2−4ac2a+−b−b2−4ac2a2=−b+b2−4ac−b−b2−4ac4a=−2b4a=−b2a$

Hence proved that, the x -coordinate of the vertex of the graph is the average of the zeros of

$f(x)=ax2+bx+c$ , $a≠0$ .

Chapter 2.1, Problem 94E

Chapter 2.2, Problem 1E

Chapter 2 Solutions

Precalculus with Limits

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Prob. 3RE Ch. 2 - Prob. 4RE Ch. 2 - Prob. 5RE Ch. 2 - Prob. 6RE Ch. 2 - Prob. 7RE Ch. 2 - Prob. 8RE Ch. 2 - Prob. 9RE Ch. 2 - Prob. 10RE Ch. 2 - Prob. 11RE Ch. 2 - Prob. 12RE Ch. 2 - Prob. 13RE Ch. 2 - Prob. 14RE Ch. 2 - Prob. 15RE Ch. 2 - Prob. 16RE Ch. 2 - Prob. 17RE Ch. 2 - Prob. 18RE Ch. 2 - Prob. 19RE Ch. 2 - Prob. 20RE Ch. 2 - Prob. 21RE Ch. 2 - Prob. 22RE Ch. 2 - Prob. 23RE Ch. 2 - Prob. 24RE Ch. 2 - Prob. 25RE Ch. 2 - Prob. 26RE Ch. 2 - Prob. 27RE Ch. 2 - Prob. 28RE Ch. 2 - Prob. 29RE Ch. 2 - Prob. 30RE Ch. 2 - Prob. 31RE Ch. 2 - Prob. 32RE Ch. 2 - Prob. 33RE Ch. 2 - Prob. 34RE Ch. 2 - Prob. 35RE Ch. 2 - Prob. 36RE Ch. 2 - Prob. 37RE Ch. 2 - Prob. 38RE Ch. 2 - Prob. 39RE Ch. 2 - Prob. 40RE Ch. 2 - Prob. 41RE Ch. 2 - Prob. 42RE Ch. 2 - Prob. 43RE Ch. 2 - Prob. 44RE Ch. 2 - Prob. 45RE Ch. 2 - Prob. 46RE Ch. 2 - Prob. 47RE Ch. 2 - Prob. 48RE Ch. 2 - Prob. 49RE Ch. 2 - Prob. 50RE Ch. 2 - Prob. 51RE Ch. 2 - Prob. 52RE Ch. 2 - Prob. 53RE Ch. 2 - Prob. 54RE Ch. 2 - Prob. 55RE Ch. 2 - Prob. 56RE Ch. 2 - Prob. 57RE Ch. 2 - Prob. 58RE Ch. 2 - Prob. 59RE Ch. 2 - Prob. 60RE Ch. 2 - Prob. 61RE Ch. 2 - Prob. 62RE Ch. 2 - Prob. 63RE Ch. 2 - Prob. 64RE Ch. 2 - Prob. 65RE Ch. 2 - Prob. 66RE Ch. 2 - Prob. 1CT Ch. 2 - Prob. 2CT Ch. 2 - Prob. 3CT Ch. 2 - Prob. 4CT Ch. 2 - Prob. 5CT Ch. 2 - Prob. 6CT Ch. 2 - Prob. 7CT Ch. 2 - Prob. 8CT Ch. 2 - Prob. 9CT Ch. 2 - Prob. 10CT Ch. 2 - Prob. 11CT Ch. 2 - Prob. 12CT Ch. 2 - Prob. 13CT Ch. 2 - Prob. 14CT Ch. 2 - Prob. 15CT Ch. 2 - Prob. 16CT Ch. 2 - Prob. 17CT Ch. 2 - Prob. 18CT Ch. 2 - Prob. 1PS Ch. 2 - Prob. 2PS Ch. 2 - Prob. 3PS Ch. 2 - Prob. 4PS Ch. 2 - Prob. 5PS Ch. 2 - Prob. 6PS Ch. 2 - Prob. 7PS Ch. 2 - Prob. 8PS Ch. 2 - Prob. 9PS Ch. 2 - Prob. 10PS Ch. 2 - Prob. 11PS Ch. 2 - Prob. 12PS Ch. 2 - Prob. 13PS

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To prove: the x -coordinate of the vertex of the graph is the average of the zeros of f ( x ) = a x 2 + b x + c , a ≠ 0 .

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