Page 1 of 3
Pre-Calculus 11
Date: _____________________
Section 4.5: Equivalent Forms of the Equation of a Quadratic Function
Goal: Complete the square to write an equation in general form as an equation in standard (vertex)
form.
Recall the VERTEX form of the quadratic equation: _________________________
Sometimes quadratic functions are not written in vertex form. Recall, that for a function to be
quadratic, we need an _________ term. Thus, quadratic functions are also written as y  ax 2  bx  c .
This is called the ___________________ form. We need to find a way to convert y  ax 2  bx  c to
y  a( x  p) 2  q . Why would we want to convert standard form to vertex form?
___________________________________________________________________________________
Before we begin…
Recall: Factor: x 2  4 x  4
x 2  12 x  36
x 2  8x  16
Exercise: From the factoring exercises above, what number must you add to make the following a
perfect square? Use algebra tiles to assist you!
Without using algebra
x 2  6 x  ______
x 2  10 x  ______
tiles…predict the value that is
needed to be added…
x 2  16 x  ______
x 2  24 x  ______
x 2  3x  ______
Completing the Square
Page 2 of 3
Pre-Calculus 11
Date: _____________________
Section 4.5: Equivalent Forms of the Equation of a Quadratic Function
Example 1: Change y  x 2  6 x  7 into vertex form by using the method of “completing the square”
Repeat again using algebra tiles. Then show how the algebra tiles correspond to the algebra
manipulation.
Extension: What is the vertex? ____________ Is it a maximum or minimum? ________
What is the max/min value? _____ At what value of “x” does this max/min occur? _____
Question: What happens when “a” (the coefficient in front of the x2 term) does not equal 1???
Example 2: Convert y  2 x 2  12 x  7 into vertex form. Use algebra tiles to show what is going
on…then write the algebra that relates to your algebra tile steps. Sketch y  2 x 2  12 x  7 .
Therefore when dealing with questions where there is an “a” value in front of the x2 term, do the
following extra step: ____________________________________________________________
Page 3 of 3
Pre-Calculus 11
Date: _____________________
Section 4.5: Equivalent Forms of the Equation of a Quadratic Function
Example 3: Convert the following into vertex form. (You try)
a) y  4 x 2  32 x  13
b) y  3x 2  21x  35
5
c) y   x 2  x  1
8
1
Example 5: Graph y   x 2  2 x  3 by first converting it to vertex form. Then state the domain and
2
range. Also find the maximum or minimum value of the function and determine what ‘x’ value does
the maximum or minimum occur.
Home Learning (McGrawHill): page 193: 2 a b d; (3, 4, 6, 7, 8) do at least 2 from each; 9, 11, 12, 16