Chapter 9.1, Problem 43PPS

a.

To determine

To decipher whether the function has a maximum or minimum value.

Answer to Problem 43PPS

The given function has a maximum value.

Explanation of Solution

Given information :

The function provided is $y=-2x^2-8x+1$ .

For the function $y=-2x^2-8x+1$ , the coefficients are $a=-2$ , $b=-8$ and $c=1$ .

Since the coefficient of ‘a’ is negative, the curve of this function is downward facing and hence the function has a maximum value.

b.

To determine

To determine the maximum or minimum value of the given function.

Answer to Problem 43PPS

The function is maximum at 9 .

Explanation of Solution

Given information :

The function provided is $y=-2x^2-8x+1$ .

Formula used :

Formula to compute the x-coordinate of the vertex is $x=-\frac{b}{2\times a}$ . Here, $a=-2$ and $b=-8$ . ‘a’ and ‘b’ are coefficients $x^2$ and $x$ respectively. The corresponding value of ‘y’ for the vertex would be the maximum or minimum.

Calculation :

Since the coefficient of ‘a’ is negative, the curve of this function is downward facing and hence the function has a maximum value.

X-coordinate of the vertex is

  $x=-\frac{b}{2\times a}$

Formula for axis of symmetry.

  $x=-\frac{-8}{2\times \left(-2\right)}$

Putting the values of ‘a’ and ‘b’ .

  $=\frac{-8}{4}$

Simplifying

  $x=-2$

Vertex can be found out by putting the value of x computed in the axis of symmetry in the original function. This will give a value of y , that will be the maximum.

  $=-2{\left(-2\right)}^2-8\left(-2\right)+1$

Putting the value of $x=-2$ in the original function.

  $=-8+16+1$

Simplifying the expression.

  $=9$

Thus, the maximum is at 9 .

c.

To determine

To calculate the domain and range for the given function.

Answer to Problem 43PPS

The domain is all real numbers, that is, $\left\{-\infty <x<\infty \right\}$ and range is all values lesser than or equal to 1 , that is, $\left\{y|y\le 9\right\}$ .

Explanation of Solution

Given information :

The function provided is $y=-2x^2-8x+1$ .

Formula used :

Formula to compute the x-coordinate of the vertex is $x=-\frac{b}{2\times a}$ . Here, $a=-2$ and $b=-8$ . ‘a’ and ‘b’ are coefficients $x^2$ and $x$ respectively. The corresponding value of ‘y’ for the vertex would be the maximum or minimum.

Calculation :

Since the coefficient of ‘a’ is negative, the curve of this function is downward facing and hence the function has a maximum value.

X-coordinate of the vertex is

  $x=-\frac{b}{2\times a}$

Formula for axis of symmetry.

  $x=-\frac{-8}{2\times \left(-2\right)}$

Putting the values of ‘a’ and ‘b’ .

  $=\frac{-8}{4}$

Simplifying

  $x=-2$

Vertex can be found out by putting the value of x computed in the axis of symmetry in the original function. This will give a value of y , that will be the maximum.

  $=-2{\left(-2\right)}^2-8\left(-2\right)+1$

Putting the value of $x=-2$ in the original function.

  $=-8+16+1$

Simplifying the expression.

  $=9$

Thus, the maximum is at 9 .

Since x can take up any real values, the range is $\left\{-\infty <x<\infty \right\}$ .

Since the domain is the values of y, it can only take up values lesser than or equal to the maximum. That is, $\left\{y|y\le 9\right\}$ .