Question

Graph f(x) =-2x^+16x-30 by factoring to find the solutions, then find the coordinates of the vertex, and the axis of symmetry. Identity the equation for the axis of symmetry, write the solutions and the coordinates of the vertex as ordered pairs. Show as much work as possible for full credit. All work should be done by hand.

Answer 1

Observe attached image

Function zeros:

(3, 0), (5, 0)

Vertex:

(4, 2)

Axis of symmetry:

`x =4`

Explanation:

First factorize the function

`f (x) = -2x ^ 2 + 16x-30`

Take -2 as a common factor.

`-2(x ^ 2 -8x +15)`

Now factor the expression
`x ^ 2 -8x +15`

You must find two numbers that when you add them, obtain the result -8 and multiplying those numbers results in 15.

These numbers are -5 and -3

Then we can factor the expression in the following way:

`f (x) = -2(x-5)(x-3)`

The quadratic function cuts the x-axis at x = 3 and at x = 5.

Now we find the coordinates of the vertex.

For a function of the form
`ax ^ 2 + bx + c` the x coordinate of its vertex is:

`x = (-b)/(2a)`

In the function
`f (x) = -2x ^ 2 + 16x-30`

`a = -2\\b = 16\\c = 30`

Then the vertice is:

`x = (-16)/(2(-2))\\\\x = 4`

The y coordinate of the symmetry axis is

`y = f (4) = -2 (4) ^ 2 +16 (4) -30\\\\y = 2`

The axis of symmetry is a vertical line that cuts the parabola in two equal halves. This axis of symmetry always passes through the vertex.

Then the axis of symmetry is the line

`x = 4`

The solutions and the vertice written as ordered pairs are:

Function zeros:

(3, 0), (5, 0)

Vertex:

(4, 2)