Algebra 2 and Trigonometry

Honors

Chapter 5: Quadratic Applications

Name:______________________________

Teacher:____________________________

Pd: _______

2

Table of Contents

Day 1: Finding the roots of quadratic equations using various methods.

SWBAT: Find the roots of a quadratic equation by completing the square, factoring, square roots, and

quadratic formula Pgs. #1 - 5

HW: pg #6-7 in packet. ##15 - 16, 19 - 20, 22 – 24,26,27

Day 2: Equations of Circles in Standard Form

SWBAT: Write the equation of a circle from standard form to center-radius form.

Pgs. #8 - 12

Hw: pg #13-15 in packet

Day 3: Solving Non-linear Systems

SWBAT: Solve non-linear system of equations

Pgs. #16 – 20

Hw: pg #21- 23 in packet

Day 4: Solving a System of Equations with 3 variables

SWBAT: Solve a system of equations with 3 variables

Pgs. #24-26

Hw: pg #27- 29 in packet

Day 5: Write equations of a parabola using a directrix and focus.

SWABT: Write an equation of a parabola using a directrix and focus.

Pgs. # 30 -36

HW: pg #37 - 39

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Day 1 - Finding the roots of quadratic equations using various methods.

Warm - Up

Analyze these four quadratic equations: x2 + 6x – 3 = 0 3x2 – 4 = 23 5x2 + 10x – 13 = 0 x2 + 2x - 24 = 0 Have each member of your group choose one equation and find the roots of the equation using one of the methods above. For the four problems below, a method can be only be used once.

Equation Method

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Quadratic Applications

Quadratic Application problems can be solved graphically and algebraically. Refer to the problem below:

1) A ball is thrown straight up at an initial velocity of 54 feet per second. The height of the ball t seconds after it is thrown is given by the formula h(t) = 54t – 12t2.

a) How many seconds after the ball is thrown will it return to the ground? b) After how many seconds does the ball reach its maximum height? c) What is the maximum height of the ball?

Graphic Solution:

a)_______________

b)_______________

c)_______________

Algebraic Solution:

a)

b)

c)

5

2)

Challenge

Solve for x.

6

Summary

7

Exit Ticket

_____________________________

_____________________________

_____________________________

_____________________________

_____________________________

_____________________________

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Day 1 – HW

#15 - 16, 19 - 20, 22 - 24

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Applications

26.

27.

10

Day 2 - Equations of Circles in Center – Radius Form.

SWBAT: Write equations and graph circles in the coordinate plane.

Warm - Up

1.

2.

Explain your answer:

Explain your answer:

11

Completing the Square of an Equation Containing Two Variables

12

Write the following equations in a) standard form and b) center-radius form.

Example 1: You Try it!

13

Practice:

Example 2: Graph the equation: x2 + y

2 - 2x + 4y - 4 = 0

Example 3: Graph: x2 + y

2 +6x -2y + 1 = 0

Example 4: Graph: x2 + y

2 +x – y – ½ = 0

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Challenge:

Summary/Closure:

Exit Ticket:

15

Day 2 - HW

Use the information provided to write the standard form equation of each circle.

1.

2.

16

3.

4.

5.

17

6. 7.

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Day 3 - Solving Non-Linear Systems

Warm – Up: Some non-linear Systems contain two variables. They are solved in the same way (substitution), but your resulting equation will have a binomial to be FOILed in the problem. Example 1: Part a: On the set of axes provided below, graph both equations.

𝑥2 + 𝑦2 = 4

𝑦 – 𝑥 = 0

Part b: What is the total number of points of

intersection of the two graphs?

Part c: Find the exact coordinates of the points of

intersection.

19

20

Example 3: Solve the system below.

are 𝒙𝟐 + 𝒚𝟐 − 𝟒 = 𝟎 and 𝒙𝟐 + 𝟒𝒙 + 𝒚𝟐 − 𝟒𝒚 + 𝟒 = 𝟎

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Example 4: Two circles whose equations are ( ) ( )x y 3 5 252 2 and ( ) ( )x y 7 5 92 2

intersect in two points. Find the exact coordinates of the points of intersection.

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Challenge A two digit number has different digits. If the difference between the square of the number and the square of the

number whose digits are interchanged is a positive perfect square, what is the two digit number?

SUMMARY Exit Ticket

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Day 3 – Homework

1. Find the intersection of the circle 2 2 29 and 3x y y x algebraically.

2. Two circles whose equations are 𝒙𝟐 − 𝟒𝒙 + 𝒚𝟐 − 𝟔𝒚 − 𝟏𝟐 = 𝟎 and 𝒙𝟐 − 𝟒𝒙 + 𝒚𝟐 − 𝟐𝒚 − 𝟒 = 𝟎

intersect in two points. Find the exact coordinates of the points of intersection.

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3.

4.

23

5.

6.

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Day 4: System of 3 Equations

Systems of equations, or more than one equation, arise frequently in mathematics. To solve a system means to

find all sets of values that simultaneously make all equations true. Of special importance are systems of linear

equations. You have solved them in your last two Common Core math courses, but we will add to their

complexity in this lesson.

Warm - Up: Solve the following system of equations by: (a) substitution and (b) by elimination.

(a) 3 2 9

2 7

x y

x y

(b) 3 2 9

2 7

x y

x y

You should be very familiar with solving two-by-two systems of linear equations (two equations and two

unknowns). In this lesson, we will extend the method of elimination to linear systems of three equations and

three unknowns. These linear systems serve as the basis for a field of math known as Linear Algebra.

Exercise #2: Consider the three-by-three system of linear equations shown below. Each equation is numbered

in this first exercise to help keep track of our manipulations.

2 15

6 3 35

4 4 14

x y z

x y z

x y z

(1)

(2)

(3)

(a) The addition property of equality allows us

to add two equations together to produce a

third valid equation. Create a system by adding

equations (1) and (2) and (1) and (3). Why is

this an effective strategy in this case?

(b) Use this new two-by-two system to solve the

three-by-three.

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Just as with two by two systems, sometimes three-by-three systems need to be manipulated by the

multiplication property of equality before we can eliminate any variables.

Exercise #3: Consider the system of equations shown below. Answer the following questions based on the

system.

4 3 6

2 4 2 38

5 7 19

x y z

x y z

x y z

Exercise #4: Solve the system of equations shown below. Show each step in your solution process.

4 2 3 23

5 3 37

2 4 27

x y z

x y z

x y z

(a) Which variable will be easiest to eliminate?

Why? Use the multiplicative property of

equality and elimination to reduce this system

to a two-by-two system.

(b) Solve the two-by-two system from (a) and find

the final solution to the three-by-three system.

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Exit Ticket/ Challenge:

Solve the following system.

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DAY 4 - SYSTEMS OF LINEAR EQUATIONS - HW

1. The sum of two numbers is 5 and the larger difference of the two numbers is 39. Find the two numbers by

setting up a system of two equations with two unknowns and solving algebraically.

2. Algebraically, find the intersection points of the two lines whose equations are shown below.

4 3 13

6 8

x y

y x

3. Show that 10, 4, and 7x y z is a solution to the system below without solving the system formally.

2 25

4 5 1

2 8 32

x y z

x y z

x y z

4. In the following system, the value of the constant c is unknown, but it is known that 8x and 4y are

the x and y values that solve this system. Determine the value of c. Show how you arrived at your answer.

5 2 3 81

1

2 35

x y z

x y z

x y cz

28

5. Solve the following system of equations. Carefully show how you arrived at your answers.

4 2 21

2 2 13

3 2 5 70

x y z

x y z

x y z

6. Algebraically solve the following system of equations. There are two variables that can be readily

eliminated, but your answers will be the same no matter which you eliminate first.

2 5 35

3 4 31

3 2 2 23

x y z

x y z

x y z

29

7. Algebraically solve the following system of equations. This system will take more manipulation because

there are no variables with coefficients equal to 1.

2 3 2 33

4 5 3 54

6 2 8 50

x y z

x y z

x y z

8.

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Day 5 – The Definition of the Parabola

Warm - Up

Patty Paper Activity

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Exercise #1: The parabola 211

4y x is shown graphed below with selected points shown. For this parabola,

its focus is the point 0, 2 and its directrix is the x-axis.

(a) How far is the turning point 0,1 from both the focus and

directrix? How far is the point 2, 2 from both?

(b) Use the distance formula to verify that the point 4, 5 is

the same distance away from the focus and directrix. Draw

line segments from the focus and directrix to this point to

visualize the distance. Repeat for the point 6,10

(c) Use the distance formula to show that the equation of this parabola is 211

4y x based on the locus

definition of a parabola.

Directrix

y

x

Focus

32

33

34

35

36

Exit ticket:

Fill in the following locus definition of a parabola with one of the words shown listed below. Words may be

used more than once.

point, line, equidistant, directrix, collection, focus

A parabola is the ____________________ of all points ______________________ from a fixed

____________________ and a fixed ______________________.

The fixed _____________________ is known as the parabola's _______________________.

The fixed _____________________ is known as the parabola's _______________________.

37

38

39

5.