4.1 The Vertex Form of a Quadratic Function
A quadratic function in vertex form, 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘, can be expressed in standard
form, 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, by expanding and simplifying. The two forms are equivalent.
What we can find out from a quadratic function expressed in vertex form, 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘:
The vertex is located at (ℎ, 𝑘)
The equation of the axis of symmetry is 𝑥 = ℎ
The function has a maximum value of 𝑘 when 𝑎 < 0
The function has a minimum value of 𝑘 when 𝑎 > 0
Example 1: For each function, state the vertex, its maximum or minimum value, its axis of
symmetry and the range. Sketch the function.
a) 𝑓(𝑥) = 3(𝑥 − 4)2 + 2
b) 𝑓(𝑥) = −(𝑥 + 2)2 − 5
Example 2: Write the equation of the quadratic function in vertex form then in standard form
a) Vertex (−2, −3) and passing through (−4, 5)
b)
Example 3: Determine the vertex and two points that satisfy the equation 𝑦 = 2(𝑥 + 3)2 − 2 and sketch the graph.
MCF 3M1: Working with Quadratic Models: Standard and Vertex Forms
Example 4: Determine the zeros of the function 𝑓(𝑥) = −2(𝑥 − 2)2 + 2.
Example 5
The path of a Bellagio water fountain is modelled by the quadratic function ℎ(𝑡) = −2(𝑡 − 3)2 + 28, where ℎ(𝑡), is in
meters and time, 𝑡, in seconds.
a) What is the maximum height the fountain reaches?
b) When does it reach the maximum height?
c) When will the fountain reach a height of 20m?
Practice: Pg. 203 # 4, 6, 7, 8ab, 9ab, 10-12
MCF 3M1: Working with Quadratic Models: Standard and Vertex Forms
4.2 Relating the Standard and Vertex Forms: Completing the Square
Previously, the quadratic functions we have been working with have been in vertex form i.e. 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘.
Unfortunately, not all quadratic functions are written in vertex form. Some quadratic functions are written in standard
form i.e. 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, while others are in intercept form 𝑓(𝑥) = 𝑎(𝑥 − 𝑟)(𝑥 − 𝑠). The process of changing the
quadratic from standard form to vertex form is called completing the square.
Example 1: Use Algebra tiles to complete the square:
a) 𝑦 = 𝑥 2 + 6𝑥 + 3
_______________
2
b) 𝑦 = 𝑥 − 4𝑥 + 1
_______________
c) 𝑦 = 2𝑥 2 + 4𝑥 + 1
_______________
Your Turn! Use Algebra Tiles to complete the square of the following quadratic relations:
Standard Form
Vertex Form
𝑦 = 𝑥 2 + 6𝑥 − 4
𝑦=
𝑦 = 𝑥 2 + 4𝑥 + 7
𝑦=
𝑦 = 2𝑥 2 + 8𝑥 + 3
𝑦=
If you look carefully at each of the questions you have used Algebra Tiles, the steps in the following algebraic approach
should be apparent:
Example 2: For each of the following quadratic relations,
i. Re-write in vertex form,
ii. Verify that you are correct by expanding
iii. State the maximum/minimum and when it occurs,
iv. Sketch the graph of the quadratic relation using the vertex.
v. State the domain and range of the relation
a) 𝑦 = 𝑥 2 − 6𝑥 + 7
MCF 3M1: Working with Quadratic Models: Standard and Vertex Forms
b) 𝑦 = 2𝑥 2 + 8𝑥 + 3
c) 𝑦 = −4𝑥 2 + 8𝑥 + 6
Beware of fractions when completing the square.
Example 3: Write 𝑦 = −2𝑥 2 + 3𝑥 + 1 in vertex form.
Example 4: Determine the values of 𝑚 and 𝑛 required to create a perfect-square trinomial.
a) 𝑥 2 − 4𝑥 = 𝑥 2 − 𝑚𝑥 + 𝑛 − 𝑛
b) 6𝑥 2 + 30𝑥 = 6(𝑥 2 + 𝑚𝑥 + 𝑛 − 𝑛)
MCF 3M1: Working with Quadratic Models: Standard and Vertex Forms
Example 5: Determine the values of 𝑎, ℎ, and 𝑘 that make the equation true.
a) 2𝑥 2 + 12𝑥 + 15 = 𝑎(𝑥 − ℎ)2 + 𝑘
b) −2𝑥 2 − 14𝑥 + 24 = 𝑎(𝑥 − ℎ)2 + 𝑘
Example 6:
A theatre sells tickets for a play. Last year it sold 500 tickets at each show for $90 per seat. The theatre’s owner is
planning to increase the price. If a recent survey showed that for every $5 increase in price, there would be a decrease
of 10 tickets sold per show,
a) What ticket price will maximize revenue?
b) What is the maximum revenue?
Practice: Pg. 213 # 2-6, 8, 10, 11, 13 For fun: 17
MCF 3M1: Working with Quadratic Models: Standard and Vertex Forms
4.3 Solving Quadratic Equations Using the Quadratic Formula
The Quadratic Formula provides us with a method of determining the roots of a quadratic equation of the form 𝑎𝑥 2 +
−𝑏±√𝑏2 −4𝑎𝑐
.
2𝑎
−𝑏−√𝑏2 −4𝑎𝑐
–𝑏+√𝑏2 −4𝑎𝑐
(
, 0) and (
, 0).
2𝑎
2𝑎
𝑏𝑥 + 𝑐 = 0. The roots of the quadratic are 𝑥 =
Graphically, the roots of the quadratic are
Since the axis of symmetry is half-way between the roots, adding the roots and
𝑏
dividing by two should result in 𝑥 = − 2𝑎 as the axis of symmetry.
Example 1:
Use the quadratic formula to solve each quadratic real equation.
a) – 𝑥 2 + 2𝑥 + 3 = 0
b) 5𝑥 2 − 8𝑥 + 1 = 0
c) 𝑥 2 − 6𝑥 + 13 = 0
MCF 3M1: Working with Quadratic Models: Standard and Vertex Forms
Example 2:
The path of a toy rocket is given by the equation ℎ(𝑥) = −0.2𝑥 2 + 8.8𝑥 + 16.2, where ℎ(𝑥) is the height in
meters and 𝑥 is the horizontal distance in meters.
a)
b)
c)
d)
e)
What is the maximum height of the rocket and at what horizontal distance did this occur?
From what height was the rocket launched?
How far did the rocket fly before hitting the ground?
What was the rocket’s height when it was 5m from the launch point?
Sketch the graph to represent the path of the rocket.
Practice: Pg. 222 # 3, 5 ad, 6-10, 12 For fun: 14
MCF 3M1: Working with Quadratic Models: Standard and Vertex Forms
4.4 Investigating the Nature of the Roots
Recall that graphically, the zeros of any parabola 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 are also the solutions of the corresponding
quadratic equation 0 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐. Therefore, the radical √𝑏 2 − 4𝑎𝑐 is of great importance and its sign plays an
important role in finding the roots. The term 𝑏 2 − 4𝑎𝑐 is called the discriminant as it discriminates the roots as follows:
If the discriminant is zero, i.e. 𝒃𝟐 − 𝟒𝒂𝒄 = 𝟎, then there is only one root. The vertex of
the corresponding parabola 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 is on the 𝒙-axis.
If the discriminant is greater than zero, i.e. 𝒃𝟐 − 𝟒𝒂𝒄 > 0, then there
are two real roots. The corresponding parabola 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 +
𝑐 crosses the 𝒙-axis at two points.
If the discriminant is less than zero, i.e. 𝒃𝟐 − 𝟒𝒂𝒄 < 0, then there are no real roots.
The corresponding parabola 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 does not cross the 𝒙-axis.
Special note: If the discriminant, 𝒃𝟐 − 𝟒𝒂𝒄 is a perfect square then the original quadratic
function was factorable.
Example 1:
Determine whether each quadratic function intersect the x-axis at one point, two points, or not at all.
a) 𝑓(𝑥) = 𝑥 2 − 2𝑥 + 3
b) 𝑓(𝑥) = −2𝑥 2 − 5𝑥 + 6
c) 𝑓(𝑥) = 3(𝑥 − 4)2 − 1
If a quadratic function is expressed in vertex form, the location of the vertex and the direction in which the parabola
opens can be used to identify how many zeros it has.
Example 2:
For what value(s) of 𝑘 does the function 𝑓(𝑥) = 𝑘𝑥 2 + 6𝑥 + 𝑘 have no zeros?
Practice: Pg. 232 # 2 abc, 4 def, 5-7, 9, 12, 14 For fun: 10
MCF 3M1: Working with Quadratic Models: Standard and Vertex Forms
4.5 Using Quadratic Function Models to Solve Problems
The vertex form of a quadratic function, 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘, can be used to solve a variety of problems, such as
determining the maximum or minimum value of the quadratic model. A quadratic equation can be solved by graphing,
factoring or using the quadratic formula.
Example 1
The population of a small town is modelled by the function 𝑃(𝑡) = 5𝑡 2 + 120𝑡 + 20 000, where 𝑃(𝑡) is the population
and 𝑡 is the time in years since 2013.
a) When will the population be 25 000?
b) What will the population be in 2038?
c) Will the population ever be 0? Explain your answer.
Example 2
The height of a football thrown in the air is given by the quadratic equation ℎ(𝑡) = −4.9(𝑡 − 2.3)2 + 28, where time, 𝑡,
is in seconds and height, ℎ(𝑡), is in metres.
a) What was the height of the football when it was thrown?
b) What is the maximum height of the ball?
c) Is the ball still in the air after 7 s? Explain.
d) When is the ball at a height of 18 m?
MCF 3M1: Working with Quadratic Models: Standard and Vertex Forms
Example 3
The function 𝑃(𝑥) = −25𝑥 2 + 2500𝑥 + 825 models the profit, 𝑃(𝑥), earned by a dance studio on the basis of the cost
of a dance lesson, 𝑥. Both profit and ticket price are in dollars.
a) What is the maximum profit of the dance studio?
b) What lesson price yields the maximum profit?
c) What is the dance studio’s profit when the cost of a lesson is $32?
Example 4
The manager of a grocery store sells a litre of milk for $2. He wants to know how much money he will earn if he
increases the price in $0.10 increments. A model of the price change is the revenue function
𝑅(𝑥) = −2𝑥 2 + 20𝑥 + 2500, where 𝑥 is the number of $0.10 increments and 𝑅(𝑥) is in dollars.
a) What is the maximum revenue?
b) What price yields the maximum revenue?
c) What is the revenue when the price of milk is $2.40?
Practice: Pg. 239 # 5-13, 17
MCF 3M1: Working with Quadratic Models: Standard and Vertex Forms
4.6 Using the Vertex Form to Create Quadratic Function Models from Data
Complete the investigation on page 242, answer questions A-K.
Example 1
Write the standard form of the quadratic equation.
a) Vertex (1, 5); y-intercept 8
b)
Example 2
Kim practises the javelin throw. The distance and height of one of the javelin throws are
given in the table.
Distance (m)
Height (m)
0
2.0
3
5.5
7
7.1
15
4.8
24
2.1
26
1.0
a) Determine the equation of a good fit in vertex form.
b) How far will the javelin travel before hitting the ground?
c) State any restrictions on the domain and range of the data.
Together: pg. 251 #7
MCF 3M1: Working with Quadratic Models: Standard and Vertex Forms
Practice: Pg. 250 # 2-6, 8-9