BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Solutions

Chapter 3.1, Problem 14E

a.

To determine

To Express: The quadratic function in standard form.

Expert Solution

Answer to Problem 14E

the quadratic function is expressed in standard form as  f(x)=1(x1)2+1

Explanation of Solution

Given: The function is f(x)=(x22x+2)

Calculation:

The quadratic function f(x)=(x22x+2) is expressed in standard form as:

  f(x)=a(xh)2+k , by completing the square. The graph of the function f is a parabola with vertex (h,k)

The parabola opens upwards if a>0 .

Solve the function:

  f(x)=(x22x+2)f(x)=1(x22x)+2                   [factor 1 from the x terms] f(x)=1(x22x+1)+211      [complete the square : add 1 inside parentheses, subtract 11 outside] f(x)=1(x1)2+1                       [factor and multiply]

On comparing the above equation with standard form f(x)=a(xh)2+k ,

Therefore, the quadratic function is expressed in standard form as  f(x)=1(x1)2+1

b.

To determine

To Find: The vertex, xintercept and yintercept

Expert Solution

Answer to Problem 14E

the vertex is (h,k)=(1,1) .

  x-intercept=0 and 2

  y-intercept=f(0)=2

Explanation of Solution

Given: The function is f(x)=(x22x+2)

Calculation:

The quadratic function f(x)=(x22x+2) is expressed in standard form as:

   f(x)=1(x1)2+1

by completing the square. The graph of the function f is a parabola with vertex (h,k) , the vertex is (h,k)=(1,1) .

The y-intercept is given when x=0 , so y-intercept=f(0)=2

The x-intercept is given when y=0 , so x22x+2=0 which implies x(x2)+2=0

Therefore, x-intercept=0 and 2

c.

To determine

To Sketch: The graph of the quadratic function.

Expert Solution

Explanation of Solution

Given: The function is f(x)=(x22x+2)

Graph:

The standard form of the function is:

   f(x)=1(x1)2+1

From the standard form it is observed that the graph is a parabola that opens upward and has vertex (1,1) . As an aid to sketching the graph, find the intercepts. The y-intercept=f(0)=2 and the x-intercept=0 and 2 . The graph f is sketched in the figure below.

Use graphing calculator to graph the function: f(x)=(x22x+2)

  Precalculus: Mathematics for Calculus - 6th Edition, Chapter 3.1, Problem 14E

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