Algebra 1 Unit 9 Quadratic Equations Name: _______________________ Part 1 Period:_________

Date Name of Lesson Notes

Tuesday 4/4

Day 1- Quadratic Transformations

Wednesday 4/5

Day 2- Vertex Form of Quadratics

Thursday 4/6

Day 3- Solving Quadratics Using Graphs

Friday 4/7

Day 4- Applying Graphs Activity

Monday 4/10

Day 5- Solving Quadratics Using Square Roots

Tuesday 4/11

Day 6- Solving Quadratics Review

Wednesday 4/12

Day 7- Quadratics Quiz

Thursday 4/13

Day 8- Vertex Form of Quadratics, Part 1

Friday 4/14

Day 9- Vertex Form of Quadratics, Part 2

Quadratic Transformations Guided Notes Day 1 Warm-up: Graph the following Absolute Value Functions and give the domain and range 1. 𝑦 = |𝑥 − 3| − 4 2. 𝑓(𝑥) = −3|𝑥 − 1| + 5

Lesson Objective: Notes

Quadratic Function: The parent functions is: ___________________________

x y

-2

-1

0

1

2

Graph: 𝑦 = |𝑥 + 2| − 4

Graph: : 𝑦 = (𝑥 + 2)2 − 4

x y

-4

-3

-2

-1

0

What do you notice?

Graph: 𝑦 = 2|𝑥| − 3

Grapℎ: 𝑦 = 2𝑥2 − 3

x y

-2

-1

0

1

2

Describe how you can determine the domain and range for both functions.

x

y

x

y

x

y

x

y

x

y Domain: Range: Describe the transformations

x

y

Domain: Range: Describe the transformations

x

y

Domain: Range: Describe the transformations

Domain: Range: Describe the transformations

Graph: 𝑦 = −1/2|𝑥 − 1| + 3

Graph: 𝑦 = −1/2(𝑥 − 1)2 + 3 x y

-1

0

1

2

3

How did think we chose the numbers for each of the tables?

How are the graphs of absolute value functions and quadratic functions the same? Different?

Summarize:

𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘 𝑎: + −

𝑎: |𝑎| > 1 |𝑎| < 1

ℎ: 𝑘:

Practice Problems

Graph and give the domain and range for each of the quadratic functions

𝑦 = 𝑥2 − 3 𝑓(𝑥) = −(𝑥 + 2)2 + 3 𝑔(𝑥) = 2(𝑥 − 2)2 − 5

Translate the graph of 𝑓(𝑥)=𝑥2 seven units to the left, six units down and vertically stretch the graph by a factor of 12.

What is the new equation?

Below is a graph for the p profits for varous selling prices of a skateboard.

x

y

Domain: Range: Describe the transformations

Domain: Range: Describe the transformations

x

y

x

y

x

y

At what selling price should you choose to make a maximum profit? At what selling prices would give you a profit of $6000? At what selling prices would you make a $0 profit?

x

y

Algebra 1 Day 2 Name_______________________________________ Vertex Form of Quadratic Functions Date_________________________Period _________ Today’s Lesson Goals:

Understand and identify the vocabulary/parts of the graph of a quadratic function.

Determine the real zeros of a quadratic function using a graph.

Determine the number of real zeros of a quadratic function using a graph.

Part 1: Graph the following quadratic function: 𝑓(𝑥) = (𝑥 − 3)2 − 4. The name for the graph of quadratic function is a ________________________________ .

Try these!

𝑓(𝑥) = (𝑥 + 3)2 − 1 𝑓(𝑥) = −2(𝑥 − 3)2 + 8

Predict Vertex:

Axis of Symmetry:

Real Zeros:

y-intercept:

Predict Vertex:

Axis of Symmetry:

Real Zeros: y-intercept:

Verify Graphically

Verify Graphically

Think about it!

Label the parts of a graph of a quadratic function: Vertex Line/Axis of Symmetry Real Zeros y-intercept

1. What relationships do you see between the following: Vertex and the Line of Symmetry Vertex and the Real Zeros Line of Symmetry and the Real Zeros Part 2: Graph the following quadratic functions

𝑓(𝑥) = −1

2(𝑥 + 1)2 − 2

𝑓(𝑥) = (𝑥 + 1)2 𝑓(𝑥) = (𝑥 − 2)2 − 4

# of real zeros:

# of real zeros:

# of real zeros:

Think about it! 1. How can you predict the number of real zeros from the equation?

Solving Quadratic Equations – Graphically day 3 Notes Warm-up:

Graph: Graph: Graph: 𝑦 = 2𝑥 − 4 𝑦 = −|𝑥 − 2| + 3 y = - 4

Solve the absolute value equation graphically 1/2|𝑥 + 2| − 3 = 𝑥 + 1 f(x) = g(x) = *Which coordinate are we wanting to find?

Objective: Notes:

Determine the number of real solutions for the following equations. 1. Graph to determine the number of solutions

(𝑥 − 2)2 + 1 = 3 𝑓(𝑥) = 𝑔(𝑥) = 2. Graph to determine the number of solutions

−2𝑥2 + 1 =1

2𝑥 + 1

𝑓(𝑥) = 𝑔(𝑥) =

3. Graph to determine the number of solutions (𝑥 − 2)2 + 2 = (𝑥 − 2)2 + 3 𝑓(𝑥) = 𝑔(𝑥) =

How many solutions do you think you can have when you solve an equation with an absolute value function and a linear function? Describe how you know this?

How many solutions does this equation have? Even though you may not be able to determine the exact solution, what would be the approximate solution(s)?

How many solutions does this equation have? Even though you may not be able to determine the exact solution, what would be the approximate solution(s)?

How many solutions does this equation have? Even though you may not be able to determine the exact solution, what would be the approximate solution(s)?

Determine the solutions for the following equations. 1. Graph to solve the equation.

𝑥2 − 3 = 1

𝑓(𝑥) = 𝑔(𝑥) = What is the solution(s)?

2. Graph to solve the equation.

−(𝑥 + 2)2 + 1 = −3

𝑓(𝑥) = 𝑔(𝑥) = What is the solution(s)?

3. Graph to solve the equation.

(𝑥 + 1)2 + 1 = −𝑥 + 2 𝑓(𝑥) = 𝑔(𝑥) =

What is the solution(s)?

4. Graph to solve the equation.

(𝑥 − 3)2 = (𝑥 − 1)2 𝑓(𝑥) = 𝑔(𝑥) =

What is the solution(s)?

Summarize:

How do you solve an equation using graphs? What are the pros for using this method? What cons for using this method? (share with your partner when you are done)

Algebra 1 Day 4 Notes Name_________________________________________ Angry Birds Applications Date_________________________Per______________

Round 1: Projectiles and Parabolas Look at the two trajectories above. 1. What is the same about the two equations?

2. What does the y-intercept represent?

3. What do the x-intercepts represent?

4. The highest part of the bird’s flight is represented by what part of the parabola?

5. Answer the following:

a.  How far does Angry Bird fly in h(x)?

b.  How high does he go?

c. How far away from the catapult is he when he is at his highest?

d. When he is 15 feet away, how high is he flying?

Round 2: Using the line of symmetry 1. When Angry Bird is 9 feet away, how high is he flying?

2. The axis of symmetry is provided.

What part of the parabola does this pass through?

What does this part represent about Angry Bird’s flight?

3. How high does the bird fly?

4. Reflect points over the axis of symmetry to complete the parabola. Do you hit any pigs?

5. How far would Angry Bird fly if he did not hit any obstacles?

6. Without solving for the whole equation: Is “a” positive or negative?

Round 3: Using the Quadratic Function 1. Angry bird wants to hit the hungry pig on the left. Angry bird and hungry pig are 18 feet away from

each other and are at same height (y-value) when angry bird is catapulted. At what distance away will

Angry Bird be the highest? Think about symmetry.

2. Now Angry Bird wants to hit the pig on the right. The equation representing his flight is:

y=−0.083(x−10.964) 2 +9.977

a) Using the picture, what is the y-intercept?

b) Using the picture, what are the x-intercepts?

c) Where is the axis of symmetry?

d) How high does Angry Bird fly (rounded to the nearest integer)?

e) Sketch the graph of Angry Bird’s flight.

Time Permitting: Write the equation for the quadratic you sketched. Use DESMOS to verify that it works. If not, keep re-writing an equation until you found the perfect (or almost perfect one).

______________________________________________________________________________________

Lesson Objective: Solving Quadratic Equations Using Square Roots day 5 Notes

Isolate the variable or expression being squared (get it ______________)

Square root both sides of the equation (include + and – on the right side!)

This means you have _____________ equations to solve!!

Solve for the variable (make sure there are no square roots in the denominator)

Simplify your square roots

1) 3x2 = 75 2.) 3x2 – 12 = 69

3.) 4x2 – 1 = 0 4.) 2

3 215

m

5.) (y + 3)2 = 49 6.) x- 2( )2

-6 = 34

7) 1

2(𝑥 − 1)2 − 8 = 0 8.) −3(𝑎 + 5)2 + 24 = 0

Can you think of another way you could solve these equations?

How solving square root equations be useful for tomorrow’s objective?

Vertex Form of Quadratic Functions – Algebraically Day 8 and 9 Notes Warm-up:

A soccer ball is kicked from ground level with an upward velocity of 90 feet per second. The equation ℎ(𝑡) = −16(𝑡 − 2.8)2 + 126.6 gives the height of the soccer ball after 𝑡 seconds.

a. What is the maximum height the ball goes? Which part of the equation tells you this?

b. How long does it take for the ball to reach its maximum height? Which part of the equation tells you this?

c. How long is the ball in the air? Justify your answer.

d. What is the height of the ball after 1 second? Show the work that gives you this answer.

e. At what times is the ball 100 feet above the ground? Show the work that gives you this answer.

Lesson Objective: Notes:

Determine the vertex, zeros, and y-intercept from vertex form of a quadratic function.

Example 1 How do you find the Zeros ? How do you find the y-Intercept ? 𝑦 = 𝑥2 − 25 Vertex: Minimum or Maximum What is the value?

Example 2 How do you find the Zeros ? How do you find the y-Intercept ? 𝑦 = −(𝑥 + 2)2 + 45 Vertex: Minimum or Maximum What is the value?

Why would finding a vertex be important in a real world example?

Why would finding the zeros be important in a real world example?

Why would finding a y-intercept be important in a real world example?

Applying vertex form to real world situations Christopher is repairing the roof on a shed. He accidently dropped a box of nails from a height of 14 feet.. This is represented by the equation ℎ(𝑡) = −16𝑡 + 14, where h is the height in feet and t is the time in seconds. How long does it take for the box of nails to hit the ground? (Do we need to find the vertex or the zeros?

The opening of a one lane tunnel shown in the graph can be modeled by the quadratic equation.

(look familiar )

𝑦=−0.18(𝑥−12.22)2+14.89

If a truck is 10 feet tall, what is the widest it could be and still fit in the tunnel?

Practice: Determine the vertex, zeros, and y-intercept of the equation: 𝑓(𝑥) = (𝑥 + 3)2 − 9 Vertex: Zeros: y-intercept:

Determine the vertex, zeros, and y-intercept of the equation: 𝑓(𝑥) = 3(𝑥 − 1)2 − 15 Vertex: Zeros: y-intercept

Della’s parents are throwing a Sweet 16 Party for her. At 10:00 a ball will slide down a pole and light up. The function

that models the drop is ℎ(𝑡) = −(𝑡 − 2.5)2 + 31.25. How long does it take for the ball to reach the bottom of the pole?