Precalculus: Mathematics for Calculus - 6th Edition
Precalculus: Mathematics for Calculus - 6th Edition
6th Edition
ISBN: 9780840068071
Author: Stewart, James, Redlin, Lothar, Watson, Saleem
Publisher: Cengage Learning
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Question
Chapter 3.1, Problem 79E
To determine

To find: The rough graph of the parabola of the function f(x)=(xm)(xn) , x -intercepts of the graph of f , x -coordinate of the vertex of parabola in terms of m and n . Confirm the answers by expanding and with the help of formulas.

Expert Solution & Answer
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Answer to Problem 79E

The graph of the parabola f(x)=(xm)(xn) is given in Figure(1) when m=1 and n=2 . The x -intercepts of the graph of f are m and n . The x -coordinate of the vertex of parabola in terms of m and n is equal to m+n2 .

Explanation of Solution

Given information:

The function is f(x)=(xm)(xn) .

Calculation:

Take a particular case when m=1 and n=2 . Sketch the graph of the function f(x)=(xm)(xn) .

  Precalculus: Mathematics for Calculus - 6th Edition, Chapter 3.1, Problem 79E

Figure(1)

By investigation of the function f(x)=(xm)(xn) with the help of graphing calculator, following observations are noted:

    Function y=(xm)(xn)mnx -intercepts of the graph y -intercept of graph Axis of symmetry
    y=(x2)(x8)282 and 816x=5
    y=(x2)(x3)232 and 36x=2.5
    y=(x3)(x5)353 and 515x=4
    y=(x+3)(x+5)353 and 515x=4
    y=(x+2)(x+4)242 and 416x=3

So, from the observations it can be concluded that the x -intercepts of the graph of f are m and n .

From the table, it is visible that the x coordinate of the vertex of parabola is m+n2 .

Now check the results by expanding.

  f(x)=(xm)(xn)=x2mxnx+mn=x2(m+n)x+mn

Factorize the function.

  f(x)=x2(m+n)x+mn=x2(m+n)x+mn+(m+n2)2(m+n2)2=(xm+n2)2+mnm2+n2+2mn4=(xm+n2)2+4mnm2n22mn4

Further simplify,

  f(x)=(xm+n2)2+4mnm2n22mn4=(xm+n2)2+m2n2+2mn4=(xm+n2)2m2+n22mn4=(xm+n2)2(mn)24

So, the x -coordinate of vertex of the parabola is m+n2 .

Now substitute f(x)=0 for the x -intercepts of parabola.

  0=(xm+n2)2(mn)24(xm+n2)2=(mn)24xm+n2=±mn2x=±mn2+m+n2

Further simplify,

  x=±mn2+m+n2x=mn2+m+n2,(mn)2+m+n2x=2m2,2n2x=m,n

So, the x -intercepts of parabola are m and n .

Therefore, the x -intercepts of the graph of f are m and n . The x -coordinate of the vertex of parabola in terms of m and n is equal to m+n2 .

Chapter 3 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

Ch. 3.1 - Prob. 11ECh. 3.1 - Prob. 12ECh. 3.1 - Prob. 13ECh. 3.1 - Prob. 14ECh. 3.1 - Prob. 15ECh. 3.1 - Prob. 16ECh. 3.1 - Prob. 17ECh. 3.1 - Prob. 18ECh. 3.1 - Prob. 19ECh. 3.1 - Prob. 20ECh. 3.1 - Prob. 21ECh. 3.1 - Prob. 22ECh. 3.1 - Prob. 23ECh. 3.1 - Prob. 24ECh. 3.1 - Prob. 25ECh. 3.1 - Prob. 26ECh. 3.1 - Prob. 27ECh. 3.1 - Prob. 28ECh. 3.1 - Prob. 29ECh. 3.1 - Prob. 30ECh. 3.1 - Prob. 31ECh. 3.1 - Prob. 32ECh. 3.1 - Prob. 33ECh. 3.1 - Prob. 34ECh. 3.1 - Prob. 35ECh. 3.1 - Prob. 36ECh. 3.1 - Prob. 37ECh. 3.1 - Prob. 38ECh. 3.1 - Prob. 39ECh. 3.1 - Prob. 40ECh. 3.1 - Prob. 41ECh. 3.1 - Prob. 42ECh. 3.1 - Prob. 43ECh. 3.1 - Prob. 44ECh. 3.1 - Prob. 45ECh. 3.1 - Prob. 46ECh. 3.1 - Prob. 47ECh. 3.1 - Prob. 48ECh. 3.1 - Prob. 49ECh. 3.1 - Prob. 50ECh. 3.1 - Prob. 51ECh. 3.1 - Prob. 52ECh. 3.1 - Prob. 53ECh. 3.1 - Prob. 54ECh. 3.1 - Prob. 55ECh. 3.1 - Prob. 56ECh. 3.1 - Prob. 57ECh. 3.1 - Prob. 58ECh. 3.1 - Prob. 59ECh. 3.1 - Prob. 60ECh. 3.1 - Prob. 61ECh. 3.1 - Prob. 62ECh. 3.1 - Height of a Ball If a ball is thrown directly...Ch. 3.1 - 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Graph the polynomial P(x) = (x + 2)3 + 27, showing...Ch. 3 - (a) Use synthetic division to find the quotient...Ch. 3 - Let P(x) = 2x3 5x2 4x + 3. (a) List all possible...Ch. 3 - Prob. 7TCh. 3 - Find all real and complex zeros of P(x) = x3 x2 ...Ch. 3 - Find the complete factorization of P(x) = x4 2x3...Ch. 3 - Find a fourth-degree polynomial with integer...Ch. 3 - Let P(x) = 2x4 7x3 + x2 18x + 3. (a) Use...Ch. 3 - Prob. 12TCh. 3 - Prob. 1PCh. 3 - Prob. 2PCh. 3 - How Fast Can You List Your Favorite Things? If you...Ch. 3 - Prob. 4PCh. 3 - Height of a Baseball A baseball is thrown upward,...Ch. 3 - Torricelli's Law Water in a tank will flow out of...
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