Quadratic Functions
Math 11 Notes
Date:
Exploring Quadratic Relations
Recall
A relation of the form y = mx + b is called a _____________ relation.
Its graph is a ___________________. The exponent of the independent
variable (x) is ________ (this is called the degree of the function).
Quadratic
A quadratic relation has the form y = ax 2  bx  c (or an
equivalent form), where a, b and c are real numbers, a ≠ 0 and x
is a variable. The degree of a quadratic relation is __________ (the
highest exponent is 2).
Relations
Think
Use the quadratic relation y = x2 to create a table of values and
a graph. What shape is the graph?
x
-2
-1
0
1
2
y
The graph of y = x2 is ________________________________________________
_________________________________________________________________________.
This shape is called a _______________________________.
Practice
Decide if the following classifications are true or false.
Function
y = 5(x+3)
y = 5(x2+3)
y = 52(x+3)
y = 5x(x+3)
y = (5x+1)(x+3)
y = 5(x+3)2 + 2
Classification
Linear
Quadratic
Quadratic
Linear
Linear
Quadratic
True/False
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Quadratic Functions
Date:
Math 11 Notes
Investigating Graphs of Quadratic Functions
Terms
Standard Form of a Quadratic Function: y = ax2 + bx + c
Quadratic term: _________
Linear term:
_________
Constant:
__________
Note: ‘a’ is the coefficient of the _________________________ term, and
‘b’ is the coefficient of the _________________________ term.
What happens to the graph of the parabola when the
coefficients of the terms (called parameters) are changed?
Activity 1
Consider the equation y = ax2. What are the values of b and c?
b = ________ and c = ________
Choose a value for ‘a’ and graph the equation. Choose the same
value for ‘a’ but make it negative and graph. What do you
notice? Now, choose a different positive ‘a’ value and graph.
What do you notice?
y = ____x2
x
y
-2
-1
0
1
2
Conclusion
y = ____x2
x
y
-2
-1
0
1
2
y = ____x2
x
y
-2
-1
0
1
2
When ‘a’ is positive, the graph ______________________________________
and has a ___________________ point. When ‘a’ is negative, the graph
_________________________________ and has a _____________________ point.
As ‘a’ gets bigger, the graph gets ___________________________________.
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Quadratic Functions
Date:
Math 11 Notes
Activity 2
Consider the equation y = x2 +bx. Choose three different values
of ‘b’ and graph the corresponding functions. What effect does
changing ‘b’ have on the graphs?
y = x2 +___x
x
y
y = x2 - ___x
x
y
y = x2+___x
x
-2
-2
-2
-1
0
1
-1
0
1
-1
0
1
2
2
2
y
Conclusion
As the value of ‘b’ increases, the graph moves _______________ (and
__________________________). The axis of symmetry shifts ____________.
As the value of ‘b’ decreases, the graph moves ______________ (and
__________________________). The axis of symmetry shifts ____________.
Activity 3
Consider the equation y = x2 + c. Choose three different values
of ‘c’ and graph the corresponding functions. What effect does
changing ‘c’ have on the graphs? What is the ‘c’ value?
y = x2 +_____
x
y
y = x2 -_____
x
y
y = x2+_____
x
-2
-2
-2
-1
0
-1
0
-1
0
1
1
1
2
2
2
Conclusion
y
As the value of ‘c’ increases, the graph moves _____________________.
As the value of ‘c’ decreases, the graph moves ____________________.
The ‘c’ value is the _____________________________________ of the graph.
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Quadratic Functions
Math 11 Notes
Date:
Characteristics of Graphs of Quadratic Functions
A parabola has a maximum y-value if the parabola opens
____________ and a minimum y-value if the parabola opens ________.
Vertex
The vertex of a parabola is the point where the curve changes
_____________________ (the minimum or maximum point). The max.
or min. of a quadratic function is the y-coordinate of the vertex.
Axis of
An axis of symmetry is a line that separates a shape into two
__________________________ parts. All quadratic parabolas have a line
of symmetry that passes through the x-coord. of the _____________.
Symmetry
Intercepts
The y-intercept is the point where the graph crosses the y-axis.
The x-intercepts are the points where the graph crosses the xaxis. There are _____________ x-intercepts depending on the graph.
Domain
The domain tells us all possible values of the independent
variable (often x).
In general, the domain is always {x | x ε R}.
Think
Why is this true? Use the graph of the parabola to help you.
Think
Give an example of a situation where this is not true (ie. what
things may ‘x’ represent that make this impossible?).
Range
The range shows possible values of the dependent variable (y)
For a parabola that opens up: {y | y ≥ min, y ε R}.
For a parabola that opens down: {y | y ≤ max, y ε R}
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Quadratic Functions
Date:
Math 11 Notes
Think
Why is this true? Use the graphs of the parabola to help you.
Example
Fill in the blanks given the following parabola:
The parabola opens _____________.
The vertex occurs at ____________.
The parabola has a _________ y – value of ______.
The eqn. of the axis of symmetry is x = ______.
The y-intercept is ______________. There are __________ x-intercepts.
The domain is {x | x ______}.The range is {y | y _____________}
Practice
Fill in the blanks given the following parabola:
The parabola opens _____________.
The vertex occurs at ____________.
The parabola has a __________ y – value of ______.
The eqn. of the axis of symmetry is x = ______.
The y-intercept is ______________. There are __________ x-intercepts.
The domain is {x | x ______}.The range is {y | y _____________}
Think
What do you notice about the points (1, 2) and (-3, 2) for the
graph above? What can you conclude?
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Quadratic Functions
Date:
Math 11 Notes
Example
The x-intercepts of the graph of a quadratic function are x = -2
and x = 4. What is the equation of the axis of symmetry?
Practice
A quadratic function has the points (2, -4) and (10, -4). What is
the equation of the axis of symmetry?
Summarizing the Characteristics of Parabolas
Summary
1) The shape of the graph is a _____________________________ (which
is a _____ - shaped ___________________).
2) The graph is symmetrical about a vertical line called the
_______________________________________________. The equation is x =
the x-coordinate of the ___________________________. Changing the
value of ________ in the function changes the axis of symmetry.
3) If a > 0, the parabola opens ____________ and the y – coordinate
of the vertex is a _____________________________. The range of the
graph is _____________________________________________________.
4) If a < 0, the parabola opens ____________ and the y – coordinate
of the vertex is a _____________________________. The range of the
graph is _____________________________________________________________.
5) The domain of the graph is _____________________________________.
6) The y-intercept of the graph is the value of ___________.
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Quadratic Functions
Date:
Math 11 Notes
Example
The function y = -3x2 + 6x – 3 has one x-intercept at x = 1.
Determine the axis of symmetry.
Think
How can you use a given function and its axis of symmetry to
determine the vertex?
Note
For any equation y = ax2 + bx + c, the equation of the axis of
symmetry (or x-coordinate of the vertex) is x = -b / 2a.
Example
Fill in the following information for the function y = -x2 – 2x + 3
The axis of symmetry is x = _______ and the vertex is _____________.
The parabola opens _________________. The y-intercept is __________.
The function has a _________________________ value of y = ___________.
The domain is ________________________________________________________.
The range is __________________________________________________________.
Practice
Complete the information for the function y = 2x2 + 4x – 6.
There are 2 x-intercepts. One at x = -3 and one at x = 1.
The axis of symmetry is x = ________. The vertex is ________________.
The parabola opens _________________. The y-intercept is __________.
The function has a __________________________ value of y = __________.
The domain is ________________________________________________________.
The range is __________________________________________________________.
Think
How many x - intercepts does each graph have?
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Quadratic Functions
Math 11 Notes
Date:
Graphing to Solve Quadratic Equations
The standard form of a quadratic equation is ax2 + bx + c = 0.
Zeroes
The zero of a quadratic function is any value of ‘x’ that makes
y = ax2 + bx + c equal to ______________.
Roots
A root is very similar to a zero. A root is any x value that makes
a quadratic equation (ax2 + bx + c = 0) _____________.
When a quadratic function is graphed, the points of the graph
where y is equal to zero (the x – intercepts) are equal to the
roots of the equation which equals the zeroes of the function.
Practice
x
-6
-5
-4
-3
-2
-1
y
The roots of the equation are
x = ________ and x = ________.
The axis of symmetry is x = ______.
The vertex is _______________.
Example
x
-0.5
0
0.5
1
1.5
2
Solve the quadratic equation by graphing: x2 + 7x + 6
y
Abed kicks a soccer ball from the ground. It follows a parabolic
path that can be modelled by the equation h = -4.9t2 + 8t where
‘h’ represents height and ‘t’ represents time. Graph the function
to answer the following:
The x – intercepts are x = _________
and x = _______. The ball hits the
ground after ________ seconds.
The maximum height of the ball is
________ m and occurs at __________s.
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Quadratic Functions
Date:
Math 11 Notes
Think
What are the constraints on the variables ‘h’ and ‘t’?
Think
What is the domain and range of the function based on the
situation given?
Domain: _________________________________________________________.
Range: ___________________________________________________________.
Practice
x
The height of a ball related to its distance travelled is modelled
by the function h = -0.1d2 + 0.2d + 1. Graph the function and
answer the following questions: h = height, d = distance
y
The y–intercept is __________. It
tells us ___________________________
___________________________________.
The x-intercepts are __________ and
__________. This means that the ball
hits the ground after ______________
______________________________________.
The vertex is __________________. This tells us ________________________
_________________________________________________________________________
_________________________________________________________________________.
The domain is ________________________________________________________.
The range is __________________________________________________________.
Practice
Calculate the height after the ball has travelled 2.3 m.
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Quadratic Functions
Math 11 Notes
Date:
Factored Form of a Quadratic Function
Recall
In grade ten, you saw that most quadratic functions can be
factored so that ax2 + bx + c = a(x – m)(x – n). For example
x2 + 5x + 4 = (x + 1)(x + 4).
Think
What are the zeroes of the quadratic function in the example?
Zero
Product
Property
The zero product property says that if the product of two real
numbers is zero, then one or both of the numbers must also be
_________________. This means that in order to find the zeroes of a
quadratic function, we need to set each of the factors equal to
zero and solve for x.
Example
Find the zeroes of the general quadratic function (x – m)(x – n).
Practice
Find the zeroes of the general quadratic function a(x - m)(x - n)
Think
What is the y – int. for the general quadratic y = a(x –m)(x –n)?
Example
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Quadratic Functions
Date:
Math 11 Notes
Example
Use the equation to sketch graphs for the quadratics below:
a) y = 2(x – 3)(x + 2)
b) y = -(x – 0.5)(x + 1)
a ___ 0, graph opens ___________
x-int: x = ______ and x = _______
axis of symmetry: _____________
vertex: ______________ max / min
a ___ 0, graph opens _________
x-int: x = _____ and x = ______
axis of symmetry: ___________
vertex: ___________ max / min
Use the x-intercepts of the graph (which are equal to the zeroes
of the function) to write the quadratic function. How can you
find the ‘a’ value?
a)
b)
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Quadratic Functions
Math 11 Notes
Date:
Solving Quadratic Equations by Factoring
Simple
Factoring
Simple factoring can be used when a = 1. Think “what two
numbers have a sum of ‘______’ and a product of ‘_____’.
Example
Use simple factoring to find the factors of y = x2 – 5x + 6
Practice
Factor the following quadratic expressions. State the x-int.
Difference
of Squares
a) y = x2 – 5x - 6
b) y = x2 + 2x
c) y = x2 + 6x + 9
d) y = -x2 + 10x – 16
Difference of squares can be used to factor when there is no
_________ term and the ax2 and c terms have opposite _____________.
1) ax2 – c = (√ax + √c)(√ax - √c)
2) -ax2 + c = (-√ax + √c)(√ax + √c)
Practice
Factor the following quadratic expressions. State the x-int.
a) y = 4x2 – 36
b) y = -9x2 + 100
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Quadratic Functions
Date:
Math 11 Notes
Example
The entry to a living room is a parabolic arch. The arch
can be modelled by the function h(t) = -0.625w2 + 5w.
Can a rectangular box that is 7 feet high and 4 feet
wide fit through the arch? Sketch the arch to help you.
Practice
State the domain and range for the function of the arch:
Domain: _______________________________________________________________
Range: _________________________________________________________________
Practice
A basketball is shot on a net. It follows a parabolic path
according to the equation h = -t2 + 4, where ‘h’ is height and ‘t’
is time. Factor to find when the ball hits the ground. Find the yint., what does it tell you?
Practice
State domain and range for the function above:
Domain: _______________________________________________________________
Range: _________________________________________________________________
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Quadratic Functions
Math 11 Notes
Date:
The Quadratic Formula
Quadratic
Formula
The quadratic formula can be used to find the zeroes of a
quadratic function when its factors are not obvious:
b  b2  4ac
x=
2a
Example
Find the roots of y = 3x2 + 5x + 2:
 ___  ___ 2  4 ______
x=
2 ____
=
x1 =
x2 =
Practice
Use the quadratic formula to find the roots of the following:
a) y = -2x2 - 3x + 2
b) y = x2 + 5x – 10
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Quadratic Functions
Math 11 Notes
c) y = 9x2 + 4x - 3
Date:
d) y = 2x2 + 6x + 1
Example
Find the zeroes of the function y = x2 + 3x + 5.
Think
What does it mean if the number under the root is negative?
Think
What does it mean if the number under the square root is zero?
Practice
Use the quadratic formula to find the x-intercepts of the graph
of the quadratic function y = 4x2 + 5x - 4. Use the x-intercepts,
vertex and direction of opening to sketch the graph.
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Quadratic Functions
Math 11 Notes
Date:
Vertex Form of a Quadratic Function
Activity
Standard Form:
y = __________________________________________
Factored Form:
y = __________________________________________
Vertex Form:
y = a(x – h)2 + k
Create a table of values to graph the function y = 2(x – 2)2 + 4.
What do you notice about ‘h’ and ‘k’?
x
-1
0
1
2
3
4
y
The ‘h’ value of a function in vertex form is ________________________
_________________________________________________________________. The ‘k’
value of a function in vertex form is ________________________________
_________________________________________________________________________.
Example
For the general function y= a(x – h)2 + k:
The graph opens _________ if a > 0, and _____________ if a < 0.
The vertex is _________________.
The maximum or minimum y point is y = _________.
The axis of symmetry is x = __________.
Practice
Find the equation of the function in vertex form from the graph:
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Quadratic Functions
Date:
Math 11 Notes
Practice
The vertex of a parabola is (-1, 2). Write an equation in vertex
form that represents all parabolas with this vertex.
Practice
If the point (-2, 4) is also on the vertex, find the value of ‘a’.
Think
Find the equations for the information below. Which form of the
quadratic should you use based on the information given?
a) The vertex is (5, -2) and there is a point (-1, -4) on the graph.
b) The factors are (x – 1) and (x + 3) and it passes through the
point (-2, 3).
Challenge
c) The quadratic has a y-intercept of 10 and passes through the
points (1, 12) and (2, 10).
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Quadratic Functions
Math 11 Notes
Date:
Solving Problems Using Quadratic Functions
Example
The daily profit for a tennis shop can be modelled by the
function P = -n2 + 240n – 5400, where ‘P’ is profit and ‘n’ is the
number of racquets produced.
a) What is the maximum profit? How many tennis racquets
must be produced?
b) What is the profit when 75 racquets are produced?
c) How many racquets are produced if the profit is $8600?
d) State the domain and range for the situation.
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Quadratic Functions
Date:
Math 11 Notes
Practice
The revenue of a ski resort is represented by y = -16x2 – 480x +
6400, where ‘y’ is the revenue in dollars and ‘x’ is the outside
temperature.
a) At what temperature does the maximum profit occur?
b) What is the revenue when the temperature is -10°C?
c) What was the temperature if the ski resort earned $8704?
d) State the domain and range.
19
Quadratic Functions
Date:
Math 11 Notes
Practice
A football follows a parabolic path when thrown. It can be
modelled by h= -4.9t2 + 24t+ 1, where h = height, t = time.
a) What was the initial height the ball was thrown from?
b) If the football is caught at a height of 1.5 m, how long was the
ball in the air?
c) If the ball was not caught, at what time would it have hit the
ground?
d) State the domain and range.
Example
A parabolic archway is 10 m wide. It is 2 m high at a point that
is 1 m away from where it touches the ground. What is the
maximum height of the archway? (Hint: Draw a diagram)
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