Chapter 4, Problem 3CRT

a.

To determine

Express $f$ in standard form.

Answer to Problem 3CRT

  $\boxed{-2{\left(x-4\right)}^2+13}$

Explanation of Solution

Given information:

Let $f$ be the quadratic function $f\left(x\right)=-2x^2+8x+5$

Express $f$ in standard form.

Calculation:

The standard form of quadratic function is,

  $f\left(x\right)=a{\left(x-h\right)}^2+k$

Now,

  $\begin{array}{l}-2x^2+8x+5=-2\left(x^2-4x\right)+5\\ =-2\left(x^2-4x+4\right)+5+8\\ =\boxed{-2{\left(x-4\right)}^2+13}\end{array}$

Hence, the solution is $\boxed{-2{\left(x-4\right)}^2+13}$

b.

To determine

Find the maximum or minimum value of $f$ .

Answer to Problem 3CRT

  $\boxed{-2{\left(x-4\right)}^2+13}$

Explanation of Solution

Given information:

Let $f$ be the quadratic function $f\left(x\right)=-2x^2+8x+5$

Find the maximum or minimum value of $f$ .

Calculation:

The standard form of quadratic function is,

  $f\left(x\right)=a{\left(x-h\right)}^2+k$

If $f$ is a quadratic function which is in the standard form, then the maximum or minimum of the function depends on the sign of $a$ .

It is observed that the sign of $a$ is negative. Here, the value of the function is maximum.

For $a<0$ , the maximum value of the function is $f\left(h\right)=k$

So the maximum value of $f$ is $\boxed{f\left(2\right)=13}$

Hence, the solution is $\boxed{f\left(2\right)=13}$

C.

To determine

Sketch the graph of $f$ .

Answer to Problem 3CRT

The solution is graph.

Explanation of Solution

Given information:

Let $f$ be the quadratic function $f\left(x\right)=-2x^2+8x+5$

Sketch the graph of $f$ .

Calculation:

The graph of the function is,

  

Hence, the solution is graph.

d.

To determine

Find the interval on which $f$ is increasing and the interval on which $f$ is decreasing.

Answer to Problem 3CRT

  $\boxed{-\infty ,2}$ , $\boxed{2,\infty }$

Explanation of Solution

Given information:

Let $f$ be the quadratic function $f\left(x\right)=-2x^2+8x+5$

Find the interval on which $f$ is increasing and the interval on which $f$ is decreasing.

Calculation:

The graph of the function is showing at $a=2$

  

The function is increasing on $\boxed{-\infty ,2}$ and decreased on $\boxed{2,\infty }$

Hence, the solution is $\boxed{-\infty ,2}$ , $\boxed{2,\infty }$

e.

To determine

How the graph of is $g\left(x\right)$ obtained from the graph of $f$ .

Answer to Problem 3CRT

The solution is graph.

Explanation of Solution

Given information:

Let $f$ be the quadratic function $f\left(x\right)=-2x^2+8x+5$

How the graph of is $g\left(x\right)=-2x^2+8x+10$ obtained from the graph of $f$ .

Calculation:

The graph of the function without $x$ is increased by $5units$ upward. The graph of $g\left(x\right)=-2x^2+8x+10$ is as shown.

  

Hence, the solution is graph.

f.

To determine

How the graph of is $h\left(x\right)$ obtained from the graph of $f$ .

Answer to Problem 3CRT

The solution is graph.

Explanation of Solution

Given information:

Let $f$ be the quadratic function $f\left(x\right)=-2x^2+8x+5$

How the graph of is $h\left(x\right)=-2{\left(x+3\right)}^2+8\left(x+3\right)+5$ obtained from the graph of $f$ .

Calculation:

The graph of the function without $x$ is increased by $3units$ left. The graph of $h\left(x\right)=-2{\left(x+3\right)}^2+8\left(x+3\right)+5$ is as shown.

  

Hence, the solution is graph.