LESSON 4: COMPLETING THE SQUARE
Learning Outcome: Learn to:



Convert quadratic functions from standard to vertex form
Analyze quadratic functions of the form 𝑦 = 𝑎𝑥² + 𝑏𝑥 + 𝑐
Write quadratic functions to model situations
As retailers consider what price to set for their items, there are many factors to take into
consideration. If items are priced too high, few people may buy them. If the prices are
set too low, sellers may not take in much revenue even though many items sell. The
key is to find the optimum price. How can you determine the price at which to sell items
that will give the maximum revenue?
Investigation:
Suppose that Grant is considering pricing for goalie masks. Last year, he sold the
goalie masks for $400 each and he sold 14 masks. He predicts that for every $40
increase in price, he will sell one fewer mask. The revenue from the goalie mask sales,
R(x) = (number of goalie masks sold)(cost per mask)
Copy and complete the table to model how Grant’s total revenue this year might change
for each price increase or decrease of $40. Continue the table to see what will happen
to the total revenue if the price continues to increase or decrease.
Number of Masks Sold
Cost per Mask ($)
Revenue ($)
16
15
320
360
5120
14
13
12
11
What pattern do you notice in the revenue as the price changes? Why do you think that
this pattern occurs?
Let x represent the number of $40 increases.

Determine an expression to represent the cost of the goalie masks

Determine an expression to represent the number of masks sold

Determine the revenue function:

Expand to give a quadratic function in standard form:
Graph the function:
 What maximum possible revenue can Grant expect?
 What price would give him the maximum possible revenue?
At which point on the graph was the maximum revenue apparent?
If we wanted to find out this information algebraically, in what form would we need the
equation to be in? Why?
Vertex form tells us more information about the graph than standard form. In order to
answer many application questions, we need to analyze the quadratic equation in
vertex form. The problem is that not every equation is given to us in this form. This is
why it is important to be able to convert a standard form equation into the vertex form.
When we complete this operation, it is called completing the square.
Completing the square involves adding a value to and subtracting a value from a
quadratic polynomial so that it contains a perfect square trinomial. You can then
rewrite this trinomial as the square of a binomial.
Ex. 𝑦 = 𝑥² − 8𝑥 + 5
Group the first two terms
𝑦 = (𝑥 2 − 8𝑥) + 5
Add and subtract the square of half the
coefficient of the x-term
𝑦 = (𝑥 2 − 8𝑥 + 16 − 16) + 5
Group the perfect square trinomial
𝑦 = (𝑥 2 − 8𝑥 + 16) − 16 + 5
𝑦 = (𝑥 − 4)2 − 11
Rewrite as the square of a binomial
The standard and vertex forms both represent the same function. The vertex form
allows to gain more information about the graph.
Converting from Standard Form to Vertex Form:
a. 𝑓(𝑥) = 𝑥² + 6𝑥 + 5
Method 1: Using Algebra Tiles
Select the tiles that coordinate with the equation:
Using the x² tile and x-tiles, create an incomplete square to represent the first two
terms. Leave the unit tiles aside for now:
To complete the square, add nine zero pairs, the nine positive tiles complete the
square and the nine negative tiles maintain equality:
Simplify the expression by removing zero pairs:
You can now express the complete the square form by reading the tiles:
𝑦 = (𝑥 + 3)2 − 4
Method 2: Algebraically:
Rewrite the function in vertex form using two methods:
𝑦 = 𝑥² + 8𝑥 − 7
Using Algebra tiles:
Using Algebra:
Complete the following Algebraically:
LESSON 5: COMPLETING THE SQUARE PART II
Learning Outcomes: Learn to:


Analyze quadratic functions of the form 𝑦 = 𝑎𝑥² + 𝑏𝑥 + 𝑐
Write quadratic functions to model situations
With a partner, define the strategies used to complete the square of a quadratic function
algebraically.
Convert 𝑦 = −3𝑥² − 27𝑥 + 13 to vertex form:
Can we complete this question another way?
How can we confirm that these two are the same?
Consider the function 𝑦 = 3𝑥² + 30𝑥 + 41
a. Convert to vertex form:
𝑏
b. Use 𝑥 = − 2𝑎 and the standard form of the quadratic function to determine the vertex.
Compare with your answer from part a.
1. 𝑦 = 2𝑥 2 − 12𝑥 + 7
2. 𝑦 = −3𝑥 2 + 12𝑥 − 15
2
3
3. 𝑦 = 3 𝑥 2 + 5 𝑥-2
Assignment: pg. 192-197 #1-12, 13, 15a, 16a