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 22E

a.

To determine

To Express: The quadratic function in standard form.

Expert Solution

Answer to Problem 22E

the quadratic function is expressed in standard form as  f(x)=6(x+1)211

Explanation of Solution

Given: The function is f(x)=(6x2+12x5)

Calculation:

The quadratic function f(x)=(6x2+12x5) 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)=(6x2+12x5) f(x)=6(x2+2x)5                        [factor 6 the x terms] f(x)=6(x2+2x+1)5(61)       [complete the square: add 1 to parentheses, subtract (61) outside] f(x)=6(x+1)211                         [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)=6(x+1)211

b.

To determine

To Find: The vertex, xintercept and yintercept

Expert Solution

Answer to Problem 22E

the vertex is (h,k)=(1,11) .

The x-intercept is 1±662

  y-intercept=f(0)=5

Explanation of Solution

Given: The function is f(x)=(6x2+12x5)

Calculation:

The quadratic function f(x)=(6x2+12x5) is expressed in standard form as:

   f(x)=6(x+1)211

by completing the square. The graph of the function f is a parabola with vertex (h,k) , the vertex is (h,k)=(1,11) .

From the standard form it is observed that the graph is a parabola that opens upward and has vertex (h,k)=(1,11) . As an aid to sketching the graph, find the intercepts.

The y-intercept=f(0)=5 . To find the x-intercepts , set f(x)=0 and factor the resulting equation.

Use Quadratic formula:

  x=b±b24ac2aHere, a=6,b=12,c=5x=12±(12)24(6)(5)2(6)x=12±144+12012x=12±26412=12±26612=6±6612=1±662

The x-intercept is 1±662

c.

To determine

To Sketch: The graph of the quadratic function.

Expert Solution

Explanation of Solution

Given: The function is f(x)=(6x2+12x5)

Graph:

The standard form of the function is:

   f(x)=6(x+1)211

From the standard form it is observed that the graph is a parabola that opens upward and has vertex (1,11) . As an aid to sketching the graph, find the intercepts.

The y-intercept=f(0)=5 and The x-intercept is 1±662 . The graph f is sketched in the figure below.

Use graphing calculator to graph the function: f(x)=(6x2+12x5)

  Precalculus: Mathematics for Calculus - 6th Edition, Chapter 3.1, Problem 22E

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