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

1 Answer
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 = (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)

enter image source here

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!