How can I find the x-intercepts and vertex of y=-2(x+1)(x-5).

Question

I don't know how to do this without graphing it. Is there any other way?

Answer 1

To find the x-intercepts, set y = 0:  -2(x+1)(x-5) = 0

 

                                                   x = -1 or x = 5

 

                                                The x-intercepts are (-1,0) and (5,0)

 

The graph is a parabola.  By symmetry, the x-coordinate of the vertex lies halfway between the x-coordinates of the x-intercepts.  

 

So, the x-coordinate of the vertex is 2.

Plug 2 into the equation of the graph to find the y-coordinate of the vertex:                          y = -2(2+1)(2-5) = 18

 

The vertex is (2, 18).   

Answer 2

First, we need to expand this out.

 

y = -2(x² - 4x - 5)

y = -2x² + 8x + 10

 

 

Now we can put this in vertex form:

 

y = a(x - h)² + k

 

where:

a is the coefficient of the x² term

Vertex has coordinates (h, k). 

 

y = -2(x² - 4x + 4) + 18

 

y = -2(x - 2)(x - 2) + 18

 

y = -2(x - 2)² + 18

 

 

Vertex = (2, 18)

 

Because the graph is a parabola that opens downward, the vertex is a maximum value.

 

 

To find the x-intercepts, set y=0 into the vertex form of the function that was just found.