Chapter 9.1, Problem 47PPS

a.

To determine

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

Answer to Problem 47PPS

The given function has a maximum value.

Explanation of Solution

Given information :

The function provided is $y=-x^2-14x-16$ .

For the function $y=-x^2-14x-16$ , the coefficients are $a=-1$ , $b=-14$ and $c=-16$ .

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 47PPS

The function is maximum at 33 .

Explanation of Solution

Given information :

The function provided is $y=-x^2-14x-16$ .

Formula used :

Formula to compute the x-coordinate of the vertex is $x=-\frac{b}{2\times a}$ . Here, $a=-1$ and $b=-14$ . ‘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.

  $x=-\frac{-14}{2\times \left(-1\right)}$

Putting the values of ‘a’ and ‘b’ .

  $=\frac{-14}{2}$

Simplifying

  $x=-7$

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.

  $=-{\left(-7\right)}^2-14\left(-7\right)-16$

Putting the value of $x=-7$ in the original function.

  $=-49+98-16$

Simplifying the expression.

  $=33$

Thus, the maximum is at 33 .

c.

To determine

To calculate the domain and range for the given function.

Answer to Problem 47PPS

The domain is all real numbers, that is, $\left\{-\infty <x<\infty \right\}$ and range is all values lesser than or equal to 33 , that is, $\left\{y|y\le 33\right\}$ .

Explanation of Solution

Given information :

The function provided is $y=-x^2-14x-16$ .

Formula used :

Formula to compute the x-coordinate of the vertex is $x=-\frac{b}{2\times a}$ . Here, $a=-1$ and $b=-14$ . ‘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.

  $x=-\frac{-14}{2\times \left(-1\right)}$

Putting the values of ‘a’ and ‘b’ .

  $=\frac{-14}{2}$

Simplifying

  $x=-7$

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.

  $=-{\left(-7\right)}^2-14\left(-7\right)-16$

Putting the value of $x=-7$ in the original function.

  $=-49+98-16$

Simplifying the expression.

  $=33$

Thus, the maximum is at 33 .

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, $\left\{y|y\le 33\right\}$ .