Question

Find the x-intercepts and the vertex of the parabola y = (x − 4)(x + 2). Find the x-intercepts of the parabola and write them as ordered pairs. Write the equation y = (x − 4)(x + 2) in standard form. With the standard form of the equation from Part II, use the quadratic formula to identify the x-value of the vertex. Substitute the x-value of the vertex from Part III into the original equation to find the y-value of the vertex. Then, write the coordinates of the vertex.

Answer 1

Given:

The eyuation of the parabola.

`y=(x-4)(x+2)`

Required:

We need to find the x-intercepts, vertex, and standard form of the equation.

Step-by-step explanation:

Set y =0 and solve for x to find the x-intercepts of the parabola.

`(x-4)(x+2)=0`

`(x-4)=0,(x+2)=0`

`x=4,x=-2`

The x-intercepts are 4 and -2.

Multipy (x-4) and (x+2) to find the stansdad form of the equation.

`y=x\left(x+2\right)-4\left(x+2\right)`

`y=(x)x+2(x)+(-4)x+(-4)2`

`y=x^2+2x-4x-8`

`y=x^2-2x-8`

The standard form of the equation is

`y=x^2-2x-8.`

which is of the fom

`y=ax^2+bx+c`

where a =1, b =-2 and c =-8.

`\text{ The x- coordinate of the vertex is }h=-(b)/(2a).`

Substitute b =-2 and a =1 in the equation.

`\text{ The x- coordinate of the vertex is }h=-((-2))/(2(1))=1`

`substitute\text{ x =1 in the equation }y=x^2-2x-8\text{ to find the y-coordinate of the vertex.}`
`y=1^2-2(1)-8=-9`

The vertex of the given parabola is (1,-9).

Final answer:

1)

The x-intercepts are 4 and -2.

2)

The standard form of the equation is

`y=x^2-2x-8.`

3)

The vertex of the given parabola is (1,-9).