Pre-Calculus 11
Ch. 3 Quadratic Functions.
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Lesson Notes 3.3: Completing the Square
Objectives:
• converting quadratic functions from standard to vertex form
• analysing quadratic functions of the form y = ax2 + bx + c
• writing quadratic functions to model situations
Up till now, you have learned how to express a quadratic function in standard form,
f (x) = ax2 + bx + c, or in vertex form, f (x) = a(x – p)2 + q. You can determine the shape of the graph
and direction of opening from the value of a in either form. The vertex form has the advantage that
you can identify the coordinates of the vertex as (p, q) directly from the algebraic form. It is useful to
be able to determine the coordinates of the vertex algebraically when using quadratic functions to
model problem situations involving maximum and minimum values.
Today, we will go over the way of converting a quadratic function in standard form to
vertex form using an algebraic process 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.
Do you remember this?
a) (x + 3)2 =
b) (x – 3)2 =
c) (x + 13)2 =
d) (x – 8)2 =
e) (x +
2 2
) =
3
f) (x –
5 2
) =
9
g) (x + m)2 =
h) (x – 2k)2 =
Now working backward.
Example 1) Rewrite each function in vertex form by completing the square.
a) f (x) = x2 + 6x + 5
b) f (x) = 3x2 – 12x – 9
c) f (x) = – 5x2 – 70x
Example 2) a) Convert the function y = 4x2 – 28x – 23 to vertex form.
b) Verify that the two forms are equivalent.
Example 3) Consider the function y = 3x2 + 12x + 8.
a) Complete the square to determine the vertex and the maximum or minimum value of the
function.
b) Use the process of completing the square to verify the relationship between the value of p
in vertex form and the values of a and b in standard form.
c) Graph the quadratic function in the grid provided.
Your Turn
Consider the function y = 2x2 – 12x + 13.
a) Complete the square to determine the vertex and the maximum or minimum value of the
function.
b) Use the process of completing the square to verify the relationship between the value of p in
vertex form and the values of a and b in standard form.
c) Graph the quadratic function in the grid provided.
Example 4) The student council at a high school is planning a fundraising event with a professional
photographer taking portraits of individuals or groups. The student council gets to charge and keep a
session fee for each individual or group photo session. Last year, they charged a $10 session fee and 400
sessions were booked. In considering what price they should charge this year, student council members
estimate that for every $1 increase in the price, they expect to have 20 fewer sessions booked.
a) Write a function to model this situation.
b) What is the maximum revenue they can expect based on these estimates. What session fee will
give that maximum?
c) How can you verify the solution?
d) What assumptions did you make in creating and using this model function?
Your Turn
A sporting goods store sells reusable sports water bottles for $8. At this price their weekly sales are
approximately 100 items. Research says that for every $2 increase in price, the manager can expect the
store to sell five fewer water bottles.
a) Represent this situation with a quadratic function.
b) Determine the maximum revenue the manager can expect based on these estimates. What
selling price will give that maximum revenue?
c) Verify your solution.