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 15E

a.

To determine

To Express: The quadratic function in standard form.

Expert Solution

Answer to Problem 15E

the quadratic function is expressed in standard form as  f(x)=1(x3)2+13

Explanation of Solution

Given: The function is f(x)=(x2+6x+4)

Calculation:

The quadratic function f(x)=(x2+6x+4) 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 downwards if a<0 .

Solve the function:

  f(x)=(x2+6x+4)f(x)=1(x26x)+4                          [factor 1 from the x terms] f(x)=1(x26x+9)+4(19)      [complete the square : add 9 inside parentheses, subtract (19) outside] f(x)=1(x3)2+13                            [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(x3)2+13

b.

To determine

To Find: The vertex, xintercept and yintercept

Expert Solution

Answer to Problem 15E

the vertex is (h,k)=(3,13) .

  x-intercept is 3±13

  y-intercept=f(0)=4

Explanation of Solution

Given: The function is f(x)=(x2+6x+4)

Calculation:

The quadratic function f(x)=(x2+6x+4) is expressed in standard form as:

   f(x)=1(x3)2+13

by completing the square. The graph of the function f is a parabola with vertex (h,k) , the vertex is (h,k)=(3,13) .

The y-intercept is given when x=0 , so y-intercept=f(0)=4

Use Quadratic formula to find the x-intercept :

  x=b±b24ac2aHere; a=1, b=6, c=4x=6±(6)24(1)42(1)x=6±36+162x=6±2132=3±13

The x-intercept is 3±13

c.

To determine

To Sketch: The graph of the quadratic function.

Expert Solution

Explanation of Solution

Given: The function is f(x)=(x2+6x+4)

Graph:

The standard form of the function is:

   f(x)=1(x3)2+13

From the standard form it is observed that the graph is a parabola that opens downward and has vertex (3,13) . As an aid to sketching the graph, find the intercepts. The y-intercept=f(0)=4 and the x-intercept is 3±13 . The graph f is sketched in the figure below.

Use graphing calculator to graph the function: f(x)=(x2+6x+4)

  Precalculus: Mathematics for Calculus - 6th Edition, Chapter 3.1, Problem 15E

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