Question

which is the graph of f(x) = (x - 1)(x + 4)?

Explanation:

Step1: Find x - intercepts

To find the x - intercepts, set \( f(x)=0 \). So, \( (x - 1)(x + 4)=0 \). Using the zero - product property, we get \( x-1 = 0 \) or \( x + 4=0 \). Solving these equations, we have \( x = 1 \) and \( x=-4 \). This means the graph of the function should cross the x - axis at \( x=-4 \) and \( x = 1 \).

Step2: Determine the direction of the parabola

First, expand the function \( f(x)=(x - 1)(x + 4)=x^{2}+4x-x - 4=x^{2}+3x - 4 \). For a quadratic function in the form \( y = ax^{2}+bx + c \), the coefficient of \( x^{2} \) (here \( a = 1 \)) determines the direction of the parabola. Since \( a=1>0 \), the parabola opens upwards.

Now, let's analyze the graphs:

• The first two graphs open downwards (since their "arms" point down), so they can be eliminated because our parabola opens upwards.
• The third graph: Let's check the x - intercepts. If we look at the third graph, the x - intercepts seem to be around \( x=-1 \) and \( x = 4 \) (not - 4 and 1), so it's not correct.
• The fourth graph: It opens upwards (since the "arms" point up) and we can check the x - intercepts. When \( x=-4 \), let's see the value of the function. Also, the x - intercepts should be at \( x=-4 \) and \( x = 1 \). The fourth graph (the bottom - left one among the four, or the last one in the given set) has x - intercepts at \( x=-4 \) and \( x = 1 \) (by looking at the grid) and opens upwards.

Answer:

The bottom - most graph (the fourth graph in the given set, with x - intercepts at \( x=-4 \) and \( x = 1 \) and opening upwards)