Question

graph of $y = (x - 4)(x + 2)$ with its vertex, axis of symmetry and $x$-labeled.

Explanation:

Step1: Find x - intercepts

To find the x - intercepts, set \(y = 0\). So, \((x - 4)(x + 2)=0\). This gives \(x-4 = 0\) or \(x + 2=0\), so \(x = 4\) and \(x=-2\). The x - intercepts are \((4,0)\) and \((-2,0)\).

Step2: Find axis of symmetry

The axis of symmetry of a parabola in factored form \(y=a(x - r_1)(x - r_2)\) is given by \(x=\frac{r_1 + r_2}{2}\). Here, \(r_1 = 4\) and \(r_2=-2\), so \(x=\frac{4+( - 2)}{2}=\frac{2}{2}=1\). So the axis of symmetry is \(x = 1\).

Step3: Find vertex

First, expand the function \(y=(x - 4)(x + 2)=x^{2}-2x - 8\). The x - coordinate of the vertex is at \(x = 1\) (from the axis of symmetry). Substitute \(x = 1\) into the function: \(y=(1)^{2}-2(1)-8=1 - 2-8=-9\). So the vertex is \((1,-9)\).

Now, let's analyze the graphs:

• The x - intercepts should be at \(x=-2\) and \(x = 4\).
• The axis of symmetry is \(x = 1\).
• The vertex is \((1,-9)\).

Looking at the four graphs, the top - left graph (assuming the first graph is top - left, second top - right, third bottom - left, fourth bottom - right) has x - intercepts at \(x = 2\) and \(x = 4\) (incorrect), the top - right has x - intercepts at \(x=-4\) and \(x = 2\) (incorrect), the bottom - left has x - intercepts at \(x=-2\) and \(x = 4\), axis of symmetry \(x = 1\) and vertex around \((1,-9)\) (matches our calculations), the bottom - right has x - intercepts at \(x=-2\) and \(x = 4\) but axis of symmetry \(x=-1\) (incorrect). So the correct graph is the bottom - left graph (or the third graph in the order).

Answer:

The bottom - left graph (or the third graph when ordered from top - left, top - right, bottom - left, bottom - right)