Chapter 6
Quadratic Functions
•
Determine the characteristics of quadratic functions•
Sketch Quadratics•
Solve problems modelled by Quadratics6.1Quadratic Functions
A quadratic function is of the form____________________________◦
where a, b and c are ______________________________________.◦
Also _________________◦
Example:
The graph of a quadratic function has a very distinct shape called a _______________________2 | P a g e
Uses of Quadratics and Parabolas
Projectile Motion
Anything that is thrown that has some horizontal motion.◦
Jumping on a bike, skis, snowboards, skidoos, etc.◦
Running off a diving board.◦
Arrows, or bullets that are shot.◦
Throwing footballs or baseballs, or kicking a soccer ball.
_________________________________________
When you take _______________________________________________ ____________________________________________________________.
The concave on a ____________________________________________
______________________________________________________________.
______________________________◦
Recall 1206 science
Parts of a Parabola
________________________◦
Place where parabola crosses ________________.
_______________________◦
Place where parabolacrosses ______________.
_______________________◦
_______________________ of a parabola
_________________________◦
The __________________ that passes through the ______4 | P a g e
Quadratic functions and Parabolas
Any relation that can be represented by a parabola can be modelled by a ___________________________________.
A quadratic function must have a _____________________________ ______________________ term has its highest ___________________.
The general equation for a quadratic is:◦
Note: In text this is called __________________________ as well!◦
_______ is the quadratic (or squared term)◦
_______ is the linear term◦
_______ is the constant term
In __ is called the _______________________________.___________________________ ___________________________ 2
Example: Which of the following is a quadratic function? If it is not state why.If it is state which direction it opens
1. 2. 3. 4. 5. 6. 7. Text Page 324 #1, 2, 4, 5, 6 3 1 y x y 2x2 x 1 2 3 2 y x x x y 6 5x2 ( 1)( 3) y x x 2 2 y x 2 3 1 y x
6 | P a g e Putting Disguised Quadratics into General Form
So, why do we need to be able to put quadratics into the general form?
So we can determine the values of a, b, and c.
These values tell us about the ___________ of the parabola.
For example, what does “a” tell us?
◦ ________________________________________________________ _______________________________________________________. What else do these values tell us?
◦ To determine this lets complete the following activity.
2
1
1
5
0
Effects of “a” on the parabola
First investigate the effect of changing the value of a
1. What happens to the direction of the opening of the quadratic if
a < 0 and a > 0?
2. A) If the quadratic opens upward, is the vertex a maximum or minimum point?
B) What if the quadratic opens downward?
3. Is the shape of the parabola effected by the parameter a ?
In other words are some graphs _____________________________? ◦ ________________________________________________________
________________________________________________________ ________________________________________________________ ◦ ________________________________________________________
_______________________________________________________. 4. What happens to the x-intercepts as the value of a is changed?
8 | P a g e Effects of “b” on the parabola
6. What is the effect of parameter b in y = ax2 + bx + c?
◦ _____________________________________________________________ _____________________________________________________________ 7. Is the parabola’s line of symmetry changing?
◦ _____________________________________________________________ Effects of “c” on the parabola
8. What is the effect of parameter c in y =ax2 +bx + c?
◦ _____________________________________________________________. 9. How can you identify the y-intercept from the equation in general form?
◦ _____________________________________________________________ 10. Is the line of symmetry affected by the parameter c?
◦ _____________________________________________________________ _____________________________________________________________
x y
Sketching Parabolas
For each of the following complete the table of values and sketch the graph on the grids provided.
1. 2 y x x y -3 -2 -1 0 1 2 3 2. 2 4 7 y x x x y
10 | P a g e -6 -5 -4 -3 -2 -1 1 -32 -28 -24 -20 -16 -12 -8 -4 x y -1 1 2 3 4 2 4 6 8 10 12 14 16 18 20 x y 3. 2 2 16 34 y x x x y -6 -5 -4 -3 -2 -1 0 Vertex: ____________ Axis of Symmetry: Domain: Range: 4. 2 3 6 10 y x x x y -1 0 1 2 3 Vertex: ___________ Axis of Symmetry: Domain: Range:
Another way of finding vertex Consider the last problem: What are the values of :
◦ a = _____ b = _____ c = _____ Calculate :
How does this value relate to the vertex?
◦ ________________________________________________________ The y-value of the vertex is found by putting the x value into the function.
Examples:
1. Find the vertex of the following:
A) (#3 in the sketches) B) 2 3 6 10 y x x 2 2 16 34 y x x 2 6 3 y x x
12 | P a g e 2. Find the maximum or minimum y-value for the following.
A) B)
3. Find the Range and the axis of symmetry for the following.
A) B) 2 3 12 1 y x x y x2 x 1 2 3 6 10 y x x y 5x2 x 21
4. The height of a soccer ball kicked into the air is given by the function:
Determine the maximum height of the ball and the time when this occurred.
5. The path of the snowboarded below is given by the equation: A) Determine the boarders max height.
5.B) Determine the domain and range the boarder’s jump.
2 5 20 y x x 2 0.15 1.28 y x x
14 | P a g e Finding vertex from the average of x-values
The x-coordinate of the vertex (___________________________) can also be found by taking average of the x-values for any two points on the parabola that has the _________________________.
Table of values
Given points
A) (3, 5) and (7,5) B) (1, 2), (0, 5), (3, 2)
x y
6.3 Drawing more accurate parabolas
A quick sketch of a parabola can be made if you know the ____________ and the ___________ the parabola _____. Example: Sketch the graph of y = x2 +2x-3
The sketch the graph of y = x2 +2x-3 can be made more _____________ if we know more ___________.
The extra points that are typically used are the places where the function ________________________________.
◦ ______________________________________ _______________
◦ To find the _____________ we set _____ and solve for __
Remember that for the function y =ax2 +bx + c the _____________ is __
x y
16 | P a g e Lets sketch the parabola
again using the
____________________ Is there another point
that we can plot based from the y-intercept? ____________________ ____________________ ____________________ ____________________ ____________________ ____________________ ____________________ ____________________ ____________________ _______________. ___________________
◦ To find the ___________ we set _____ and solve for __
◦ __________________
◦ When finding the _______________ of a function you are actually finding the ____________ for the function.
ie. You are finding _____________ that
make the function _________________. ◦ To solve for __ we will need to ____________ the
quadratic.
You will be required to use _______________ developed in Mathematics 1201 to determine the ________.
x y
x y
SO, how do we factor _________________ ◦ Product and Sum Method
◦ You must find two numbers such that the product of the two numbers equals ___, the ______________. ◦ The sum of the two numbers must equal ___, the
________________.
So now what?
To find the zeros from the __________________ of the function we use the ____________________________ The ____________________________ states that:
◦ if the ____________ of two real numbers is ____, ◦ then ____ or ______ of the numbers must be ___. ◦ Find the zeros of ___________________
Lets sketch the parabola again using the _______________ as well
18 | P a g e
E
E
x
x
a
a
m
m
p
p
l
l
e
e
s
s
For each of the following find A) the vertex
B) the y-intercept C) the x-intercepts
Then use this information to draw a graph of the function. 1.
y
x
25
x
4
x y
2. 2
2 4 30
y x x
x y
20 | P a g e 3. x y 2 3 13 4 y x x
1201– Factoring Trinomials of the form ax2+bx+c
Warm UP Factor:
When the leading coefficient of a trinomial is not 1 the
_________________________________________________ _________________________________________________ _______________________.
When factoring the trinomial ax2+bx+c, we will find two numbers
that multiply to give the product ______and will have a sum of
___
Factor 2x2+11x+12
2
5 6
22 | P a g e Find the x- intercepts of 6x2+17x+10
Note** Before starting to factor a trinomial, always check to see if you can remove a _____________________.
IV Solve 6k2-11k-35 = 0
24 | P a g e VI Solve 3s2-13s-10 = 0
Practice for Assignment 6.2
Sketch the graph of the following by finding the intercepts and vertex of parabola. A) B) 2 6 7 y x x x y 2 3 12 y x x x y
26 | P a g e C) D)
(
2)(
4)
y
x
x
x y1
(
2)(
4)
3
y
x
x
Note:
The last two equations in your notes and on Assignment 6.2 were expressed in a special form called the ____________________ of the quadratic.
Factored form of a quadratic
________________________________
◦ Where r and s are the ___________ of the function
______________________ are _______________ of the parabola.
◦ __ is the ________________________________. Example
Sketch the graph of a parabola that passes through the points (-3,0) and (4, 0)
Make _________ more parabolas that are
different from the first one, but still have
____________________
How many parabolas do you think are possible? Explain
x y
28 | P a g e Note:
The goal is for you to recognize that a __________ of parabolas are possible when the ________________________________.
◦ For example the factored form ____________________ represents the family of parabolas that we have drawn through the points (-3,0) and (4, 0)
When provided with an _______________, however, you can narrow down the __________ formula for the quadratic equation.
In order to determine the multiplier _____________________ in the factored form y = a(x +3)(x - 4), you need to choose ____ point on the parabola and use substitution.
Select one of the parabolas that you have drawn and determine the leading coefficient “a”.
Write the factored form for your parabola.
Expand the factored form to express the quadratic in standard form.
x y
Example:
Determine the quadratic function, in standard form, with factors (x + 3) and (x - 5) and a y-intercept of -5.
30 | P a g e x
y
x y
Find the equations of the parabola graphed below: A)
Each form of a quadratic has its own characteristics,
and its own benefits.
If the quadratic is written in general or standard form,
________________________, you can determine:
◦
the
_______________
and the direction of the
______________ of the parabola directly from the
equation.
◦
the x-coordinate of the ____________________
by using
If the equation is written in factored form,
_____________________ you can determine:
◦
the
__________________
of the graph
◦
and the direction of the ______________ of the
parabola.
◦
the x-coordinate of the vertex by taking the
___________________________________
Both of these forms required ___________________
to find the ______________.
There is one more form of the quadratic which enables
you to _____________ determine the vertex called the
___________________________,
32 | P a g e
Vertex Form of a Quadratic
•
Vertex Form : ____________________________
•
“a” is the ________________________.
–
If “a” is positive the parabola opens _____
–
If “a” is negative the parabola opens ____
•
The point ________ is the _________ of the
parabola.
Example: What is the vertex of the following?
A)
B)
Sketch:
What is the vertex?
Which way is the
graph opened?
3
)
1
(
2
x
2y
( 2) 5 3 2 2 x y 21
1
2
2
y
x
Sketch:
What is the vertex?
Which way is the graph
opened?
What is the equation of
the axis of symmetry?
What is the max/min
y-value?
What is the domain and
range?
2
2
1
3
34 | P a g e
Finding Equations of Parabolas
•
What is the vertex?
•
Write the vertex form for
this parabola
•
How do we find a?
What is the equation of the axis of symmetry?
What is the max/min y-value? What is the domain/range?
Quiz
36 | P a g e
Word Problems
1. A ball is thrown from an initial height of 1 m and follows a parabolic path. After 2 seconds, the ball reaches a maximum height of 21 m. Algebraically determine the quadratic function that models the path followed by the ball, and use it to determine the approximate height of the ball at 3 seconds.
A)How is the shape of the graph connected to the situation?
B) What do the coordinates of the vertex represent?
C) What do the x and y-intercepts represent?
2. The goalkeeper kicked the soccer ball from the ground. It reached a maximum height of 24.2 m after 2.2 seconds. The ball was in the air for 4.4 s.
A) Determine the quadratic function that models the height of the ball above the ground.
B) How high is the ball after 4 s?
38 | P a g e 3. A quarterback throws the ball from an initial height of 6 feet. It is caught by the receiver 50 feet away, at a height of 6 feet. The ball reaches a maximum height of 20 feet during its flight. Determine the quadratic function which models this situation and state the domain and range.
Distinguishing between a function and a function that
represents a real world situation
4B) Suppose the function h(t) = -0.15t2 + 6t represents the height of a
ball, in metres, above the ground as a function of time, in seconds. State the domain and range.
5. The path of a model rocket can be described by the quadratic
function y = - x2 - 12x, where y represents the height of the rocket, in
metres, at time x seconds after takeoff.
A) Identify the maximum height reached by the rocket and determine the time at which the rocket reached its maximum height.
40 | P a g e
Other Word Problems
6. You have 600 meters of fencing and a large field. You want to make a rectangular enclosure split into two equal lots. What dimensions would yield an enclosure with the largest area?
7. John has ordered 40 feet of fencing to build a dog enclosure. One side of his house will be one of the sides of the enclosure. Determine the dimensions that will give his dogs the maximum area to run around in.
Will it Fit?
8. A two lane highway runs through a tunnel that is framed by a
parabolic arch, which is 20 m wide. The roof of the tunnel, measured 4 m from the right base is 4 m above the ground. Can a truck that is 4 m wide and 6.2 high pass through the tunnel?