Question

Given the following quadratic functions: g ( x ) = − x 2 − x + 6 g(x)=−x 2 −x+6 and n ( x ) , n(x), shown in the table below. Which statement about these functions is true? x x n ( x ) n(x) − 3 −3 − 7 −7 − 2 −2 0 0 − 1 −1 5 5 0 0 8 8 1 1 9 9 2 2 8 8 3 3 5 5 4 4 0 0 5 5 − 7 −7 Over the interval -1≤x≤1, the average rate of change for n(x) is less than that for g(x) The y-intercept of g(x) is greater than the y-intercept for n(x) The function g(x) has a greater maximum value than n(x) The sum of the roots of n(x)=0 is greater than the sum of the roots of g(x)

Answer 1

Functions can be represented on a table or as an equation. The true statement is that:

n(x) has a greater sum of roots

Given

$g(x) = x^2 - x + 6$

$\left[\begin{array}{cccccccccc}x&-3&-2&-1&0&1&2&3&4&5&n(x)&-7&0&5&8&9&8&5&0&-7\end{array}\right]$

Start by testing the options:

(a) The average rate of change over the interval $-1 \le x \le 1$

Average rate is calculated as follows:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

For g(x),

$m = \frac{g(-1) - g(1)}{-1 - 1}$

$m = \frac{g(-1) - g(1)}{-2}$

Calculate g(-1) and g(1)

$g(x) = x^2 - x + 6$

$g(-1) =(-1)^2 -(-1) + 6 = 8$

$g(1) =1^2 -1 + 6 = 6$

So, we have:

$m = \frac{8 - 6}{-2}$

$m = \frac{2}{-2}$

$m = -2$

For n(x),

$m = \frac{n(-1) - n(1)}{-2}$

$m = \frac{5 - 9}{-2}$

$m = \frac{- 4}{-2}$

$m = 2$

Hence, n(x) has a greater rate of change over $-1 \le x \le 1$

Option (a) is false

(b) y-intercepts

This is where $x = 0$

For g(x),

$g(x) = x^2 - x + 6$

$g(0) = 0^2 - 0 + 6 = 6$

For n(x),

$n(0) = 8$

n(x) has a greater y-intercept.

Option (b) is false

(c) Greater maximum

For g(x),

$g(x) = x^2 - x + 6$

Differentiate

$g'(x) = 2x - 1$

Equate to 0

$2x -1= 0$

Solve for x

$2x =1$

$x = \frac 12$

$x = 0.5$

So, the maximum of g(x) is:

$g(x) = x^2 - x + 6$

$g(0.5) = 0.5^2 - 0.5 + 6$

$g(0.5) = 5.75$

The maximum of n(x) is at:

$n(1) = 9$

n(x) has a greater maximum.

Option (c) is false

(d) Greater sum of roots

Since other options are false,

Option (d) will be true

Hence, n(x) has a greater sum of roots

Read more about tables and equations at:

https://brainly.com/question/13140019