Question

which is the graph of $f(x) = (x - 1)(x + 4)$? four graphs of parabolas on coordinate grids with options to select

Explanation:

Step 1: Find the x - intercepts

To find the x - intercepts of the function \(f(x)=(x - 1)(x + 4)\), we set \(f(x) = 0\). So, \((x-1)(x + 4)=0\). Using the zero - product property, if \(ab = 0\), then either \(a = 0\) or \(b = 0\). So, \(x-1=0\) gives \(x = 1\) and \(x + 4=0\) gives \(x=-4\). So the graph of the function should cross the x - axis at \(x=-4\) and \(x = 1\).

Step 2: Determine the direction of the parabola

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 \(x^{2}\) (i.e., \(a\)) determines the direction of the parabola. Here, \(a = 1>0\), so the parabola opens upwards.

Step 3: Analyze the options

• The first two graphs open downwards (since the coefficient of \(x^{2}\) would be negative for a downward - opening parabola), so we can eliminate them.
• The third graph: Let's check the x - intercepts. The x - intercepts of the third graph seem to be at non - 1 and non - 4 values. Wait, no, let's re - check. Wait, the fourth graph: Wait, no, let's go back. Wait, the function has x - intercepts at \(x=-4\) and \(x = 1\). Let's check the fourth graph. The fourth graph crosses the x - axis at \(x=-4\) (or around there) and \(x = 1\) (or around there) and opens upwards. The third graph: Let's see, the third graph's x - intercepts are not at - 4 and 1. Wait, maybe I made a mistake. Wait, the function \(f(x)=(x - 1)(x + 4)\) has x - intercepts at \(x = 1\) and \(x=-4\). Let's check the graphs:
• First graph: Opens downward (so \(a<0\)), but our function has \(a = 1>0\), so eliminate.
• Second graph: Opens downward ( \(a<0\)), eliminate.
• Third graph: Let's check the x - intercepts. It seems to cross the x - axis at, say, \(x=-1\) and \(x = 4\) (not - 4 and 1), so eliminate.
• Fourth graph: Opens upward (\(a>0\)) and crosses the x - axis at \(x=-4\) and \(x = 1\) (approximately).

Answer:

The fourth graph (the one at the bottom left, the last graph) is the graph of \(f(x)=(x - 1)(x + 4)\).