Chapter 9.1, Problem 48PPS

a.

To determine

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

Answer to Problem 48PPS

The given function has a minimum value.

Explanation of Solution

Given information :

The function provided is $y=4x^2+40x+44$ .

For the function $y=4x^2+40x+44$ , the coefficients are $a=4$ , $b=40$ and $c=44$ .

Since the coefficient of ‘a’ is positive, the curve of this function is upward facing and hence the function has a minimum value.

b.

To determine

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

Answer to Problem 48PPS

The function is minimum at -56 .

Explanation of Solution

Given information :

The function provided is $y=4x^2+40x+44$ .

Formula used :

Formula to compute the x-coordinate of the vertex is $x=-\frac{b}{2\times a}$ . Here, $a=4$ and $b=40$ . ‘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 positive, the curve of this function is upward facing and hence the function has a minimum value.

X-coordinate of the vertex is

  $x=-\frac{b}{2\times a}$

Formula for axis of symmetry.

  $x=-\frac{40}{2\times \left(4\right)}$

Putting the values of ‘a’ and ‘b’ .

  $=\frac{-40}{8}$

Simplifying

  $x=-5$

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 minimum.

  $=4{\left(-5\right)}^2+40\left(-5\right)+44$

Putting the value of $x=-5$ in the original function.

  $=100-200+44$

Simplifying the expression.

  $=-56$

Thus, the minimum is at -56 .

c.

To determine

To calculate the domain and range for the given function.

Answer to Problem 48PPS

The domain is all real numbers, that is, $\left\{-\infty <x<\infty \right\}$ and range is all values lesser than or equal to -56 , that is, $\left\{y|y\ge -56\right\}$ .

Explanation of Solution

Given information :

The function provided is $y=4x^2+40x+44$ .

Formula used :

Formula to compute the x-coordinate of the vertex is $x=-\frac{b}{2\times a}$ . Here, $a=4$ and $b=40$ . ‘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 positive, the curve of this function is upward facing and hence the function has a minimum value.

X-coordinate of the vertex is

  $x=-\frac{b}{2\times a}$

Formula for axis of symmetry.

  $x=-\frac{40}{2\times \left(4\right)}$

Putting the values of ‘a’ and ‘b’ .

  $=\frac{-40}{8}$

Simplifying

  $x=-5$

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 minimum.

  $=4{\left(-5\right)}^2+40\left(-5\right)+44$

Putting the value of $x=-5$ in the original function.

  $=100-200+44$

Simplifying the expression.

  $=-56$

Thus, the minimum is at -56 .

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 greater than or equal to the minimum. That is, $\left\{y|y\ge -56\right\}$ .