Given the function:
[tex]f\mleft(x\mright)=\mleft(x+2\mright)\mleft(x-4\mright)[/tex]
You can rewrite it as follows:
[tex]y=\mleft(x+2\mright)\mleft(x-4\mright)[/tex]
You need to remember that the y-value is zero when the function intersects the x-axis. Then, you need to make it equal to zero, in order to find the x-intercepts:
[tex]\begin{gathered} 0=\mleft(x+2\mright)\mleft(x-4\mright) \\ (x+2)(x-4)=0 \end{gathered}[/tex]
Solving for "x", you get these two values:
[tex]\begin{gathered} x+2=0\Rightarrow x_1=-2 \\ \\ x-4=0\Rightarrow x_2=4 \end{gathered}[/tex]
In order to find the vertex, you can follow these steps:
1. Find the x-coordinate of the vertex with this formula:
[tex]x=-\frac{b}{2a}[/tex]
To find the value of "a" and "b", you need to multiply the binomials of the equation using the FOIL Method. This states that:
[tex]\mleft(a+b\mright)\mleft(c+d\mright)=ac+ad+bc+bd[/tex]
Then, in this case, you get:
[tex]\begin{gathered} y=(x)(x)-(x)(4)+(2)(x)-(2)(4) \\ y=x^2-4x+2x-8 \end{gathered}[/tex]
Add the like terms:
[tex]y=x^2-2x-8[/tex]
Notice that, in this case:
[tex]\begin{gathered} a=1 \\ b=-2 \end{gathered}[/tex]
Then, you can substitute values into the formula and find the x-coordinate of the vertex of the parabola:
[tex]x=-\frac{(-2)}{2\cdot1}=-\frac{(-2)}{2}=1[/tex]
2. Substitute that value of "x" into the function and then evaluate, in order to find the y-coordinate of the vertex:
[tex]\begin{gathered} y=x^2-2x-8 \\ y=(1)^2-2(1)-8 \\ y=1-2-8 \\ y=-9 \end{gathered}[/tex]
Therefore, the vertex of the parabola is:
[tex](1,-9)[/tex]
Knowing the x-intercepts and the vertex of the parabola, you can graph it.
Hence, the answer is:
