QUADRATIC RELATIONS
AND
FUNCTIONS
CHAPTER 13
NOTES PACKET 2009 - 10
SOLVING QUADRATIC EQUATIONS (DAY 1)
QUADRATIC EQUATION:
General Quadratic Equation:
STANDARD FORM of a quadratic equation:
HOW TO SOLVE QUADRATIC EQUATIONS:
Step 1:
Write equation in Standard Form.
Step 2:
Factor the quadratic equation
Step 3:
After the problem has been factored we will complete a step
called the “T” step. Create a T separating the two ( ).
Step 4:
Once ( ) are separated, set each ( ) = to 0 and solve for the
variable.
Step 5:
Check each of the roots in the ORIGINAL quadratic equation
EXAMPLES: Solve each equation. Check the roots.
1.
Find the roots: r 2  12r  35  0
2.
2
Solve for y: y 2  11y  24  0
x 2  5x  6  0
3.
Find the roots:
4.
Solve for y: y 2  3y  28
5.
Find the roots:
6.
Find the roots: 3x 2  3x  7x  3
x 2  x  30
3
SOLVING QUADRATIC EQUATIONS (DAY 2)
RECALL SOLVING STEPS:
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
INCOMPLETE (SPECIAL) QUADRATICS:
ex.
ax 2  bx  0
and
ex.
ax 2  c  0
Solve the following equations. Check all the roots:
1.
z2  4  0
2.
a 2  36  0
3.
x 2  4x
4.
x 2  8x  0
5.
3 x 2  12  0
6.
5y 2  45
4
CHALLENGE PROBLEMS: Solve and Check the roots
7.
Solve for x:
x(x  3)  40
8.
Solve for x:
x
3

5 x2
9.
Solve for x:
3x x 2

4
8
5
QUADRATIC WORD PROBLEMS
CONSECUTIVE INTEGERS (DAY 3)
Review of Consecutive Integers “Let Statements”
Integers
Consecutive Integers
Consecutive Even Integers
Consecutive Odd Integers
EXAMPLES:
1.
When the square of a certain number is diminished by 9 times the number
the result is 36. Find the number.
2.
A certain number added to its square is 30. Find the number.
3.
The square of a number exceeds the number by 72. Find the number.
4.
Find two positive numbers whose ratio is 2:3 and whose product is 600.
6
5.
The product of two consecutive odd integers is 99. Find the integers.
6.
Find two consecutive positive integers such that the square of the first is
decreased by 17 equals 4 times the second.
7.
The ages of three family children can be expressed as consecutive
integers. The square of the age of the youngest child is 4 more than eight
times the age of the oldest child. Find the ages of the three children.
8.
Find three consecutive odd integers such that the square of the first
increased the product of the other two is 224.
7
GEOMETRIC WORD PROBLEMS (DAY 4)
Remember:
 Draw a picture and Write a “Let” statement
 Write an equation
 Solve the equation ( REMEMBER: YOU CAN’T HAVE A NEGATIVE LENGTH)
 Check to see if your solution makes sense
 Re-Read the problem to make sure you answered the question
1. The ratio of the measures of the base and the altitude of a parallelogram is
3:4. The area of the parallelogram is 1,200 square centimeters. Find the
measure of the base and altitude of the parallelogram.
2. The altitude of a triangle is 5 less than its base. The area of the triangle is 42
square inches. Find its base and altitude.
3. The length of a rectangle exceeds its width by 4 inches. Find the dimensions
of the rectangle it its area is 96 square inches.
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4. If the measure of one side of a square is increased by 2 centimeters and the
measure of the adjacent side is decreased by 2 centimeters, the area of the
resulting rectangle is 32 square centimeters. Find the measure of one side of
the square.
5. Joe’s rectangular garden is 6 meters long and 4 meters wide. He wishes to
double the area of his garden by increasing its length and width by the same
amount. Find the number of meters by which each dimension must be
increased.
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MORE GEOMETRIC PROBLEMS (DAY 5)
1.
In a trapezoid, the smaller base is 3 more than the height, the larger base
is 5 less than 3 times the height, and the area of the trapezoid is 45 square
centimeters. Find, in centimeters, the height of the trapezoid.
2.
If the length of one side of a square is tripled and the length of an
adjacent side is increased by 10, the resulting rectangle has an area that
is 6 times the area of the original square. Find the length of a side of the
original square.
3.
The length of a rectangle is 7 units more than its width. If the width is
doubled and the length is increased by 2, the area is increased by 42
square units. Find the dimensions of the original rectangle.
4.
The side of one square is 2 centimeters longer than the side of the second
square. If the sum of their areas is 100cm2, find the length of the side of
each square.
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THE GRAPH OF A QUADRATIC FUNCTION
(DAY 6)

The Quadratic Equation is written as: _______________________ , this
equation has a degree of _________.

Where a, b and c are integer coefficients (where a

0)

The graph of this equation is called a ________________, it is _______________

Parabolas are functions because they _________________________________
2 TYPES OF PARABOLA SHAPES
When “a” is positive, the parabola opens: _______
Where the curve reaches a ___________________
When “a” is negative, the parabola opens: _______
Where the curve reaches a ___________________
Draw in the line of symmetry of the parabola on the grid.
y
x
This line of symmetry is called the __________________________
 It is always a vertical line that goes through the turning point
of the curve.
Formula:
Axis of Symmetry:
Turning Point: Is another term for the vertex of the parabola. The
“vertex” has the coordinates of x, y  .
To Find Turning Point (T.P.)


y
Roots of the equation are the points where the parabola
x
_____________________
the x – axis, so y = ___________,
What are the roots of the parabola on the grid to the left? _________
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GRAPHING QUADRATIC FUNCTIONS
How to Graph Parabolas:
1. Find the axis of symmetry by using the formula.
2. Substitute the x value back into the
equation to find the turning point and describe it as a max or min pt.
3. Make a table of values.
4. Graph the points.
EX1: GRAPH: y  x 2  4x
EX2: GRAPH: y   x 2  2x  3 (  4  x  2 )  This
is called an interval, which means your table should cover
the x values of -4 to 2.
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EXPLORING THE GRAPHED
QUADRATIC EQUATION (DAY 7)
Quadratic functions are written in the form: ______________________________
The x – intercepts (when x = 0) of the parabola y  ax2  bx  c are called the
_________________ of the equation ( ax2  bx  c  0 )
How many roots are possible to obtain from a quadratic equation?
Draw a picture to illustrate each situation
y
y
x
y
x
x
EX1. Given the following graph of the equation y = x2 – 7x + 10. Answer the
following questions.
What is the axis of symmetry? __________
What are the coordinates of the turning
point?________
Is the T.P. a max or minimum point? ________
How many roots are there?__________
What are the solutions of this equation? _____
What are the solutions called? ________
Now, solve the equation:
0 = x2 – 7x +10
What do you notice?
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EX2. GRAPH:
y  x 2  6x  5
What is the axis of symmetry? __________
What are the coordinates of the turning
point?_______
Is the T.P. a max or minimum point? ________
How many roots are there?__________
What are the solutions of this equation? ________
What do you call these solutions? __________
Now try to solve the above equation algebraically! How do we do this?
Write an equation for each graph below.
Ex.3
Ex.4
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SOLVING QUADRATIC-LINEAR SYSTEMS (DAY 8)
Two equations will be given to you with the directions to solve the system
graphically.
 One equation will be a quadratic. This equation has degree ________
 The second equation will be linear. This equation has degree ________
You will have to create a table of values for the quadratic equation and graph
the linear equation using ______________.
Where the two graphs ___________________, this is your ______________________.
There are three possible situations as answers illustrated below. Indicate the
number of solutions in each representation.
Examples:
1.
Solve the following system of equations graphically and check.
y   x 2  4x  3
x  y 1
To check on your graphing calculator (find intersection):
1)
Go to 2nd
2)
Move cursor to wanted intersection point
Trace
(Calculate) and pick
15
#
5
Enter
(intersection)
Enter
Enter
2.
Solve the following system of equations graphically and check.
y  x 2  4x  4
y  2 x  4
3.
The graphs of the equations y  x 2 and x  2 intersect in:
(1)
4.
1 point
(2)
2 points
(3)
3 points
(4)
4 points
(4)
(6, 0)
Which is a solution or the following system of equations?
y  2 x  15
y  x 2  6x
(1)
5.
(3, –9)
(2)
(0, 0)
(3)
(5, 5)
When the graphs of the equations y  x 2  5x  6 and x  y  6 are drawn
on the same set of axes, at which point do the graphs intersect?
(1)
(4, 2)
(2)
(5, 1)
(3)
16
(3, 3)
(4)
(2, 4)
SOLVING QUADACTIC –LINEAR SYSTEMS
ALGEBRAICALLY (DAY 9)
Two equations will be given to you with the directions to solve the system
algebraically

One equation will be a quadratic. This equation has degree ________

The second equation will be linear. This equation has degree ________
Steps:
1. Make sure both equation are in y = form if necessary
2. Substitute the linear equation into the ‘y part’ of the quadratic equation, to
have only one variable left to solve in the equation.
3. Get NEW quadratic equation into standard form (_________________________)
and______________________________
4. Since it is a quadratic: Must FACTOR TO SOLVE FOR X.
(How many answers should you get?_______)
5. Must find other variable (y) by substituting your ‘x’ answers into one of the
equation and solve for y.
6. Check solutions
Examples:
1. Solve the following system:
y  x2  x  2
y  2x
17
2. Find the solutions of:
3. Solve for the solutions:
y  x 2  4x  3
x  y 1
y  x 2  7x  13
xy 2
18
APPLICATIONS WITH PARABOLIC FUNCTIONS (DAY 10)
Using the graph at the right, It shows the height h
in feet of a small rocket t seconds after it is launched.
The path of the rocket is given by the equation:
h = -16t2 + 128t.
1. How long is the rocket in the air? _________
h (height (feet))
250
200
2. What is the greatest height the rocket reaches? ____
3. About how high is the rocket after 1 second? _______
4. After 2 seconds,
a. about how high is the rocket?_________
b. is the rocket going up or going down? ________
5. After 6 seconds,
a. about how high is the rocket? _______
b. is the rocket going up or going down? ________
150
100
50
1
2
3
4
5
6
time (seconds)
6. Do you think the rocket is traveling faster from 0 to 1 second or from 3 to 4
seconds? Explain your answer.
7. Using the equation, find the exact value of the height of the rocket at 2
seconds.
8. A ball is thrown in the air. The path of the ball is represented by the equation
h = -t2 + 8t. Graph the equation over the interval 0  t  8 on the
height (meters)
accompanying grid.
What is the maximum height of the ball?_______________
How long is the ball above 7 meter? ________________
time (seconds)
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7
8
QUADRATIC APPLICATION WORD PROBLEMS (ALGEBRICALLY)
(DAY 11)
Things to remember when completing quadratic application word problems:
t is______. It represents __.
h or d is _________/distance. It represents __.
When an object hits the ground (water), its height = 0.
1.
After t seconds, a ball tossed in the air from the ground level reaches a
height of h feet given by the equation h = 144t – 16t2.
a.
What is the height of the ball after 3 second?
b.
What is the maximum height the ball will reach?
c.
Find the number of seconds the ball is in the air when it reaches a
height of 224 feet.
d.
After how many seconds will the ball hit the ground before
rebound?
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2.
A rocket carrying fireworks is launched from a hill 80 feet above a lake.
The rocket will fall into lake after exploding at its maximum height. The
rocket’s height above the surface of the lake is given by
h = -16t2 + 64t + 80.
a.
What is the height of the rocket after 1.5 second?
b.
What is the maximum height reached by the rocket?
c.
How long will it take for the rocket to hit 128 feet?
d.
After how many seconds after it is launched will the rocket hit the
lake?
3.
A rock is thrown from the top of a tall building. The distance, in feet,
between the rock and the ground t seconds after it is thrown is given by
d = -16t2 – 4t + 382. How long after the rock is thrown is it 370 feet from the
ground?
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