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Introduction to Quadratics: standard form, axis of
symmetry, and vertex
Content Objective: I can identify key features of a quadratic
equation (direction of opening, vertex, and axis of
symmetry). Language Objective: I can explain how to identify
the key features of a quadratic function.
WHAT IS A QUADRATIC EQUATION?
STANDARD FORM
STANDARD FORM OF A QUADRATIC
EQUATION:
GRAPH
When graphed, a quadratic equation creates a U shaped
curve called a
___________________________________________________
TYPES OF PARABOLAS
Use your graphing calculator to sketch the following:
● If “a” is a _________________________, the
parabola opens ______ like a
smile. ☺ ● If “a” is a _________________________, the
parabola opens ______ like a
frown. ☹
AXIS OF SYMMETRY VERTEX
KEY FEATURES
_______________________________________
_______________________________________
FORMULA FOR THE AXIS OF SYMMETRY:
_______________________________________
● When the vertex is the lowest point, it is called a
___________________
● When the vertex is the highest point it is called a
___________________
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FIND THE AXIS OF SYMMETRY AND VERTEX, THEN SKETCH EACH
PARABOLA
1. x 5y = x2 + 8 + 1
Axis of Symmetry Vertex Sketch
2. − 0x 3y = x2 + 1 − 2
Axis of Symmetry Vertex Sketch
3. x 2xy = 3 2 − 1 + 5
Axis of Symmetry Vertex Sketch
4. − x 4x 2y = 3 2 − 2 − 4
Axis of Symmetry Vertex Sketch
5. − xy = x2 + 4
Axis of Symmetry Vertex Sketch
6. y = x2 − 3
Axis of Symmetry Vertex Sketch
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Graphing quadratic equations Content Objective: I can graph
quadratic equations in standard form. I can identify
key features of a quadratic function. Language Objective: I
can explain how to graph and identify key features of quadratic
functions.
EXAMPLES
Graph each quadratic equation using a table. Identify the axis of
symmetry, vertex, domain, and range.
1. y = x2 Axis of
Symmetry: Vertex: Domain: Range:
2. x y = x2 + 2 − 1 Axis of
Symmetry: Vertex: Domain: Range:
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3. − x 7 y = x2 − 8 − 1 Axis of
Symmetry: Vertex: Domain: Range:
4. − x x y = 2 2 + 4 + 1 Axis of
Symmetry: Vertex: Domain: Range:
5. x 1 y = x2 − 6 + 1 Axis of
Symmetry: Vertex: Domain: Range:
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6. − y = x2 − 2 Axis of
Symmetry: Vertex: Domain: Range:
7. x x y = 2 2 + 8 Axis of
Symmetry: Vertex: Domain: Range:
8. − x y = x2 + 4 + 3 Axis of
Symmetry: Vertex: Domain: Range:
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9. − x y = x2 − 2 Axis of
Symmetry: Vertex: Domain: Range:
10. − x 8x 0 y = 3 2 − 1 − 2 Axis of
Symmetry: Vertex: Domain: Range:
11. x x 1y = 21 2 − 6 + 1
Axis of
Symmetry: Vertex: Domain: Range:
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Vertex Form of a quadratic equation Content Objective: I
can graph a quadratic equation in vertex form. Language
Objective: I can explain how to graph an equation in vertex
form.
● Vertex form of a Quadratic
Equation:
● _____________ is the vertex; _____________ is the axis of
symmetry.
Directions: Give the axis of symmetry
and vertex of each equation.
1. x )y = ( + 4 2 − 2
Axis of Symmetry
___________________ Vertex
___________________
2. − x )y = ( − 3 2 Axis of Symmetry
___________________ Vertex
___________________
3. − xy = 2 2 + 3 Axis of Symmetry
____________________ Vertex
____________________
EXAMPLES Graph each quadratic equation using a table. Identify
the axis of symmetry, vertex, domain, and
range.
4. − x ) y = ( + 2 2 + 7 Axis of
Symmetry: Vertex: Domain: Range:
5. (x ) y = 3 − 1 2 Axis of
Symmetry: Vertex: Domain: Range:
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● The most simplistic quadratic equation is
__________________________
● This is known as the
______________________________________________
● A transformation is a _________________ to the
__________________ or
______________________ of a figure.
EXAMPLES Graph each quadratic equation using a table.
Identify the axis of symmetry, vertex, domain, and
range.
6. x ) y = ( + 2 2
Transformations:
7. y = x2 + 5
Transformations:
8. x ) y = ( + 1 2 − 6
Transformations:
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9. − x ) y = ( − 4 2 + 1
Transformations:
10. x y = 3 2 − 7
Transformations:
11. − (x )y = 21 − 3 2 − 2
Transformations:
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PUT IT TOGETHER
Given a quadratic equation in vertex form, (x )y = a − h 2 + 5
● is the _______________ shift.h
● is the _______________ shift.k
● If is negative, the function is _________________ across the
___ - ______a
● represents a vertical __________________.a| | > 1
● represents a vertical _______________.0 < a| | < 1
(x ) y = a − h 2 + k
Directions: Transformations from the
parent function are= y x2 described below. Write an
equation to represent the function.
12. Translated 2 units right.
13. Translated 5 units up
14. Translated 3 units left and 4 units
down
15. Translated 7 units right and 4 units
up
16. Reflected over the , then translatedxisx − a
3 units down.
17. Reflected over the , then translatedxisx − a
5 units right and 2 units down.
18. Vertically compressed by a factor of ,
then3
1 translated 8 units up.
19. Vertically stretched by a factor of 2, reflected
over the
, then translatedxisx − a 4 units left.
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QUADRATIC ROOTS AND THE DISCRIMINANT Content
Objective: I can find the number of zeros/roots/solutions of a
quadratic function using the discriminant. I can
find exact zeros/roots/solutions using a graph. Language
Objective: I can explain how to find the number of solutions and
exact
solutions of a quadratic.
1.
2.
3.
EXAMPLES Find the solutions of the following quadratic equations
by graphing.
1. x y = x2 + 4 − 5
Solution(s):
_________________________
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2. x y = x2 − 2 + 1
Solution(s):
_________________________
3. x y = − x2 + 2 − 3
Solution(s):
_________________________
4. 0x 6 y = x2 − 1 + 1
Solution(s):
_________________________
5. − y = x2 + 9
Solution(s):
_________________________
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6. − x y = 3x2 + 6
Solution(s):
_________________________
Formula:
● If , then there are ____ solutions.d > 0
● If , then there are ____ solutions.d = 0
● If , then there are ____ solutions.d < 0
Directions: Use the discriminant
to determine the number of solution.
7. xy = x2 + 5 + 4
8. x 0y = x2 − 3 + 1
9. 0x 5y = x2 + 1 + 2
10. x xy = 2 2 − 4 − 3 11. x xy = 2 2 − 4 − 3
12. − x xy = 3 2 + 5 − 8
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Solving quadratics by factoring (Day 1) Content Objective:
I can find solution of quadratic functions by
factoring. Language Objective: I can explain how to find
solutions of quadratics functions by
factoring.
In many cases we can find the solutions (or roots, zeros,
x-intercepts) of a quadratic equation by factoring,
rather than graphing.
Follow the steps below to find the solutions of the given
equation by factoring.
EXAMPLES Solve the following quadratic equations by
factoring.
1. x x2 + 4 + 3 = 0
2. 1x 4 x2 + 1 + 2 = 0
3. x2 + x − 2 = 0
4. x 7 x2 + 6 − 2 = 0
5. 0x 1 x2 − 1 + 2 = 0
6. 0 x2 − x − 2 = 0
7. 0x 5 x2 + 1 + 2 = 0
8. x 6 x2 − 8 + 1 = 0
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9. x x2 − 8 = 0
10. x 5x 3 2 + 1 = 0
11. x 2x 6 2 − 1 = 0
12. x x 8 2 − 6 = 0
13. 4 x2 − 6 = 0 14. 5 x2 − 2 = 0
15. x 1 4 2 − 8 = 0
16. x 9 9 2 − 4 = 0
17. x 1x2 + 4 = 2
18. 5 xx2 − 4 = 4
19. x 4 x x2 − 5 − 6 = 7
20. 0x 9 x x2 − 1 + 4 = 4 + 1
21. 1x x1 2 = x2 + 8
22. 6x 1 2 = 9
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Solving quadratics by factoring (Day 2) Content Objective:
I can find solution of quadratic functions by
factoring. Language Objective: I can explain how to find
solutions of quadratics functions by
factoring.
EXAMPLE 1 x x 2 3 2 + 9 − 1 = 0
EXAMPLE 2 x 0x 0 5 2 − 2 − 6 = 0
EXAMPLE 3 x x2 2 + 3 − 5 = 0
EXAMPLE 4 x 2x8 2 − 2 + 5 = 0
EXAMPLES Solve the following quadratic equations by
factoring.
1. x 0x 2 2 + 1 + 8 = 0
2. x 4x 8 4 2 − 2 − 2 = 0
3. x 3x 0 3 2 + 1 − 1 = 0
4. x x 5 2 − 8 + 3 = 0
5. x 3x 2 2 + 1 − 7 = 0
6. x x 6 2 + 5 + 1 = 0
7. x x 4 2 − 8 − 5 = 0
8. x 2x 4 2 + 1 + 9 = 0
9. x x − 3 2 + 7 = 2
10. x x 5 2 + 7 = x2 + 2
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Solving quadratics by Square Roots Content Objective: I can
find solutions to quadratics by using square roots. Language
Objective: I can explain how to find solutions of quadratics using
square roots.
Quadratic equations of the form
______________________________
(no “bx” term!) can be solve using square roots!
**REMEMBER THAT A POSITIVE NUMBER ALWAYS HAS TWO SQUARE
ROOTS!**
EXAMPLES Solve the following quadratic equations by square
roots.
1. 6 x2 − 1 = 0
2. 00 x2 − 1 = 0
3. 5 x2 + 2 = 0
4. 8 x2 + 7 = 8
5. − x2 − 5 = 4
6. x 4 6 2 = 5
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7. x − 8 − 2 2 = 9
8. x 843 2 = 4
9. x 08 3 2 − 1 = 0
10. x 5 5 2 − 4 = 0
11. x 6 9 2 − 1 = 0
12. 5x 0 6 2 2 + 1 = 4
13. x 6 7 2 + 6 = 3 14. x 521 2 + 3 = 7
15. x 5 2 − 1 = x2
16. 6x 4 5 1 2 − 3 = 1
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ZEROS CAN BE IRRATIONAL! Solve the following quadratic
equations. Write all answers in simplest radical form.
17. x2 − 3 = 0
18. x2 − 8 = 0
19. 6 x2 + 8 = 5
20. 6 9 x2 − 1 = 5
21. x 26 2 2 − 1 = 0
22. x − 28 − 3 2 = 2
23. 0 8− x2 − 1 = 1
24. x 1 521 2 − 1 = 2
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Solving quadratics by the quadratic formula Content
Objective: I can use the quadratic formula to find solutions to
quadratic functions. Language Objective: I
can explain how to use the quadratic formula to find solutions
to quadratic functions.
The quadratic formula is another method
to use to solve a quadratic equation.
Solve the equation below using the quadratic
formula.
EXAMPLES Solve each equation using the quadratic
formula
1. x 0 x2 − 8 = 2
2. x x 2 2 2 + 7 + 3 = 1
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3. x 2 3 2 − 1 = 0
4. 5x x x2 + 1 = 6
5. 0x 1 − x2 − 1 − 2 = 0
6. x x 2x 4 2 + 9 = 1
7. x 0 x2 + 7 = x − 1
8. x x x 3 2 − 5 = 4 − 3 2
REMEMBER: ZEROS CAN BE IRRATIONAL! Solve the following
quadratic equations. Write all answers in simplest radical
form.
9. xx2 + 4 + 1 = 0
10. x − x2 − 2 + 7 = 0
11. xx2 + 3 = 8 − x2
12. x x − 4 2 − 7 = 2
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Area problems Content Objective: I can solve quadratic
equation word problems involving area and consecutive
integers. Language Objective: I can explain how to solve area
and consecutive integer word problems.
1. Given the diagram below, find the value of if
the area of the rectangle is 78x square
meters.
2. Given the diagram below, find the dimensions of the
rectangle if the area of the rectangle is 108 square
meters.
3. Find the dimensions of the rectangle below if the
area is 128 square feet.
4. The dimensions of a rectangle can be expressed as and .
If the areax + 3 x − 8 of the rectangle is 60 square
inches, find the value of 60.
5. The length of a rectangular garden is 4 meters
more than its width. The area of the rectangle is 60
meters. Find the dimensions of the
garden.
6. The length of a rectangle is 6 meters less than
its width. Find the dimensions of the rectangle if its
area is 27 square meters.
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7. A square is altered so that one dimension is increased by 5
inches and the other dimension is decreased by 2
inches. If the area of the resulting rectangle is 98 square inches,
find the area of the original
square.
8. Given the diagram to the left, if the area of the shaded
region is 59 square inches, what are the dimensions of
the outside
rectangle?
9. The product of two positive consecutive integers is
56. Find the
integers.
10. The product of two positive consecutive even integers
is 80. Find the integers.
11. Find two positive consecutive odd integers such that
the square of the first, added to 3 times
the second, is
24.
12. Find three positive consecutive integers such that the
product of the second integer and the third integer is
72.
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PROJECTILE MOTION PROBLEMS Content Objective: I can solve
projectile motion problems. Language Objective: I can
explain how to solve projectile motion
problems.
1. A soccer ball is kicked from the ground with an initial
upward velocity of 90 feet per second. The equation
gives the height of the ball after seconds.− 6t 0t h = 1 2 + 9 h t
a. Find the maximum height of the
ball. b. How many seconds will it take
for the ball to reach the ground?
2. An apple is launched directly upward at 64 feet per second
from an 80-foot tall platform. The equation for this
apple’s height at time seconds after launch is .h t − 6t 4t 0 h = 1
2 + 6 + 8
a. Find the maximum height of the
apple. b. How many seconds will it
take for the apple to reach the
ground?
3. In science class, the students were asked to create a
container to hold an egg. They would then drop this container
from a window that is 25 feet above the ground. If
the equation of the container’s pathway can be modeled by the
equation , how− 6t 5 h = 1 2 + 2 long will it take the
container to reach the ground?
a. Find the maximum height of the
container. b. How many seconds will it
take for the container to reach the
ground?
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4. A penny is dropped off the Empire State Building, which is
1,250 feet tall. If the penny’s pathway can be modeled by the
equation , how long would it take the− 6t 250 h = 1 2 + 1
penny to strike a 6-foot tall
person?
5. Some fireworks are fired vertically into the air from the
ground at an initial speed of 80 feet per second. The
equation for this object’s height at time seconds after launch ish
t . How long will it take the fireworks to reach the
ground?− 6t 0t h = 1 2 + 8
6. The Apollo’s Chariot, a roller coaster at Busch Gardens,
moves at 110 feet per second. The equation of the ride
can be represented by the equation .− 6t 01t 0 h = 1 2 + 1 + 1
What is the maximum height reached by this
ride?
7. Eva is jumping on a trampoline. Her height at time can be
modeled by the equationh t . Would Eva reach a height
of 14 feet?− 6t 0t h = 1 2 + 2 + 6
8. An astronaut on the Moon throws a baseball upward with an
initial velocity of 10 meters per second, letting go of
the baseball 2 meters above the ground. The equation of
the baseball pathway can be modeled by . The same
experiment is done on− .8t 0th = 0 2 + 1 + 2 Earth, in
which the pathway is modeled by equation .− .9t 0t h = 4 2 + 1 + 2
How much longer would the ball stay in the air on the
Moon compared to on
Earth?
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