Question

How do you find the vertex and the intercepts for `y = (x − 3)(4x + 2)`?

Answer 1

You must first write the function in standard form, by distributing, and then complete the square into vertex form.

Explanation:

Before we do what was mentioned above, we can determine the y intercept, as well as the x intercepts.

`0 = (x - 3)(4x + 2)`

There will therefore be x intercepts at `(3, 0)` and at `(-1/2, 0)`.

`y = (0 - 3)(4(0) + 2)`

`y = -3(2)`

`y = -6`

The y intercept is at `(0, -6)`.

Now for the vertex:

Completing the square is a process for converting quadratic functions from standard form (`y = ax^2 + bx + c`) to vertex form (`y = a(x - p)^2 + q`)

`y = 4x^2 - 10x - 6`

`y = 4(x^2 - 5/2x + n) - 6 ->` factoring out the 4. "n" is the value that will turn the expression in parentheses into a perfect square.

`n = (b/2)^2`

`n = ((-5/2)/2)^2`

`n = 25/16`

`y = 4(x^2 - 5/2x + 25/16 - 25/16) - 6 ->` adding and subtracting the value of n inside the parentheses, in order to keep the expression equivalent.

`y = 4(x - 5/4)^2 - 25/4 - 6 ->` extracting the negative value from the parentheses. This needs to be multiplied with parameter a in vertex form.

`y = 4(x - 5/4)^2 - 49/4`

In vertex form, `y = a(x - p)^2 + q`, the vertex is located at `(p, q)`. Hence, our vertex is at `(5/4, -49/4)`

Here is the graph of this function:

graph{y = (x - 3)(4x + 2) [-40, 40, -20, 20]}

Practice exercises:

1. Determine the vertex and intercepts of the following functions:

a) `y = (x + 1)(x - 6)`

b) `y = (-2x - 5)(1/4x + 3)`

1. Use the following graph of `y = f(x)` to answer questions a), b), c) and d)

a) What is the vertex of this function?

b) What are the x intercepts of this function?

c) Challenging!! What is this graph's equation?

d) Challenging!! Use the equation of the graph to find the coordinates of the y intercept.

Hopefully this helps!