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

May 30, 2016

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.

$0 = \left(x - 3\right) \left(4 x + 2\right)$

There will therefore be x intercepts at $\left(3 , 0\right)$ and at $\left(- \frac{1}{2} , 0\right)$.

$y = \left(0 - 3\right) \left(4 \left(0\right) + 2\right)$

$y = - 3 \left(2\right)$

$y = - 6$

The y intercept is at $\left(0 , - 6\right)$.

Now for the vertex:

Completing the square is a process for converting quadratic functions from standard form ($y = a {x}^{2} + b x + c$) to vertex form ($y = a {\left(x - p\right)}^{2} + q$)

$y = 4 {x}^{2} - 10 x - 6$

$y = 4 \left({x}^{2} - \frac{5}{2} x + n\right) - 6 \to$ factoring out the 4. "n" is the value that will turn the expression in parentheses into a perfect square.

$n = {\left(\frac{b}{2}\right)}^{2}$

$n = {\left(\frac{- \frac{5}{2}}{2}\right)}^{2}$

$n = \frac{25}{16}$

$y = 4 \left({x}^{2} - \frac{5}{2} x + \frac{25}{16} - \frac{25}{16}\right) - 6 \to$ adding and subtracting the value of n inside the parentheses, in order to keep the expression equivalent.

$y = 4 {\left(x - \frac{5}{4}\right)}^{2} - \frac{25}{4} - 6 \to$ extracting the negative value from the parentheses. This needs to be multiplied with parameter a in vertex form.

$y = 4 {\left(x - \frac{5}{4}\right)}^{2} - \frac{49}{4}$

In vertex form, $y = a {\left(x - p\right)}^{2} + q$, the vertex is located at $\left(p , q\right)$. Hence, our vertex is at $\left(\frac{5}{4} , - \frac{49}{4}\right)$

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) $y = \left(x + 1\right) \left(x - 6\right)$

b) $y = \left(- 2 x - 5\right) \left(\frac{1}{4} x + 3\right)$

1. Use the following graph of $y = f \left(x\right)$ 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!