The graph of the function [tex]f(x)=(x+2)(x-4)[/tex] is shown.

Which describes all of the values for which the graph is negative and increasing?
A. all real values of [tex]x[/tex] where [tex]x\ \textless \ -2[/tex]
B. all real values of [tex]x[/tex] where [tex]-2\ \textless \ x\ \textless \ 4[/tex]
C. all real values of [tex]x[/tex] where [tex]1\ \textless \ x\ \textless \ 4[/tex]
D. all real values of [tex]x[/tex] where [tex]x\ \textless \ 0[/tex]

Find the roots of the function f ( x ) = ( x + 2 ) ( x − 4 ) to determine where the graph is negative: − 2 < x < 4 .
Determine the vertex of the parabola to find where the graph is increasing: 1"> x > 1 .
Find the intersection of the intervals where the graph is negative and increasing.
The graph is negative and increasing for 1 < x < 4 , so the answer is 1 < x < 4 ​ .

Explanation

Analyzing the function We are given the function f ( x ) = ( x + 2 ) ( x − 4 ) and its graph. We need to find the interval where the graph is negative and increasing. First, let's analyze the function.

Expanding the function The function f ( x ) is a quadratic function, which can be expanded as f ( x ) = x 2 − 2 x − 8 . This is a parabola that opens upwards because the coefficient of the x 2 term is positive (1).

Finding where the graph is negative The graph is negative between the roots of the function. The roots are the values of x for which f ( x ) = 0 . Thus, we have ( x + 2 ) ( x − 4 ) = 0 , which gives us the roots x = − 2 and x = 4 . Therefore, the graph is negative for − 2 < x < 4 .

Finding where the graph is increasing To find where the graph is increasing, we need to find the vertex of the parabola. The x-coordinate of the vertex is given by x v ​ = − b / ( 2 a ) , where a and b are the coefficients of the quadratic equation f ( x ) = a x 2 + b x + c . In our case, a = 1 and b = − 2 , so x v ​ = − ( − 2 ) / ( 2 ∗ 1 ) = 1 . The vertex is at x = 1 . Since the parabola opens upwards, the graph is increasing for 1"> x > 1 .

Finding the intersection Now we need to find the interval where the graph is both negative and increasing. We know the graph is negative for − 2 < x < 4 and increasing for 1"> x > 1 . The intersection of these two intervals is 1 < x < 4 .

Final Answer Therefore, the graph is negative and increasing for all real values of x where 1 < x < 4 .


Examples
Understanding where a function is negative and increasing is useful in many real-world scenarios. For example, imagine a company's profit is modeled by the function f ( x ) = ( x + 2 ) ( x − 4 ) , where x represents the number of units sold. The company wants to know the range of sales where they are making a loss (negative profit) but their profit is increasing. This corresponds to the interval where the function is negative and increasing. Knowing this interval ( 1 < x < 4 ) helps the company make informed decisions about production and sales strategies to move towards profitability.

Answered by GinnyAnswer | 2025-07-04

The function f ( x ) = ( x + 2 ) ( x − 4 ) is negative between its roots − 2 and 4 , specifically for values − 2 < x < 4 . The graph is increasing for values greater than 1 . Therefore, it is negative and increasing for values in the interval 1 < x < 4 , making option C the correct answer.
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Answered by Anonymous | 2025-07-06