Question

non core algebra i b - cr
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 the vertex's x - coordinate

The function is \(f(x)=(x - 4)(x + 1)=x^{2}-3x - 4\). For a quadratic function \(y = ax^{2}+bx + c\), the x - coordinate of the vertex is given by \(x=-\frac{b}{2a}\). Here, \(a = 1\), \(b=-3\), so \(x =-\frac{-3}{2\times1}=\frac{3}{2}=1.5\). The parabola opens upwards (since \(a = 1>0\)).

Step2: Analyze the increasing/decreasing intervals

A parabola that opens upwards decreases to the left of the vertex (\(x<1.5\)) and increases to the right of the vertex (\(x > 1.5\)). Let's check each option:

• Option 1: The function is not increasing for \(x < 0\) (since for \(x<1.5\) it's decreasing), so this is false.
• Option 2: The roots are \(x = 4\) and \(x=-1\), but the vertex is at \(x = 1.5\), so the increasing intervals are \(x>1.5\), not \(x < - 1\) or \(x>4\), so this is false.
• Option 3: The decreasing interval is \(x < 1.5\), not \(-1
• Option 4: Since the parabola opens upwards and the vertex is at \(x = 1.5\), the function is decreasing for all \(x<1.5\), so this is true.

Answer:

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