Introduction to Quadratics: standard form, axis of...

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 ___________________

1

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


3. x 2xy = 3 2 − 1 + 5


4. − x 4x 2y = 3 2 − 2 − 4


5. − xy = x2 + 4


6. y = x2 − 3


2

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:

3

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:

4

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:

5

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:

6

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

____________________


4. − x ) y = ( + 2 2 + 7 Axis of Symmetry: Vertex: Domain: Range:

5. (x ) y = 3 − 1 2 Axis of Symmetry: Vertex: Domain: Range:

7

● The most simplistic quadratic equation is __________________________

● This is known as the ______________________________________________

● A transformation is a _________________ to the __________________ or

______________________ of a figure.


6. x ) y = ( + 2 2 Transformations:

7. y = x2 + 5 Transformations:

8. x ) y = ( + 1 2 − 6 Transformations:

8

9. − x ) y = ( − 4 2 + 1 Transformations:

10. x y = 3 2 − 7 Transformations:

11. − (x )y = 21 − 3 2 − 2

Transformations:

9

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.

10

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): _________________________

11

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): _________________________

12

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

13

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

14

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

15

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

16

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

17

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

18

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

19

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

20

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

21

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.

22

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.

23

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?

24

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?

25

Introduction to Quadratics: standard form, axis of...

Documents

Transcript of Introduction to Quadratics: standard form, axis of...