Question

the graph of the function ( f(x) = (x - 4)(x + 1) ) is shown below. which statement about the function is true?

• the function is increasing for all real values of ( x ) where ( x < 0 ).
• the function is increasing for all real values of ( x ) where ( x < -1 ) and where ( x > 4 ).
• the function is decreasing for all real values of ( x ) where ( -1 < x < 4 ).
• the function is decreasing for all real values of ( x ) where ( x < 1.5 ).

Explanation:

Step1: Find vertex x-coordinate

First, expand the function:
$$f(x)=(x-4)(x+1)=x^2-3x-4$$
For quadratic $ax^2+bx+c$, vertex x is $-\frac{b}{2a}$. Here $a=1, b=-3$, so:
$$x=-\frac{-3}{2(1)}=1.5$$

Step2: Analyze increasing/decreasing

Since $a=1>0$, the parabola opens upward. This means:

• The function decreases left of the vertex ($x<1.5$)
• The function increases right of the vertex ($x>1.5$)

Step3: Match to options

Compare with given choices: only the statement about decreasing for $x<1.5$ is true.

Answer:

The function is decreasing for all real values of x where x < 1.5.