Day 1 (90min) Day 2 (90min) Day 3 (90min) Day 4 (90min) Day 5 (90min)

Objectives: SWBAT

1) sketch a graph of a quadratic equation 2) determine the number of roots to a quadratic equation from the graph 3) determine domain and range of quadratic equations 4) Predict the effect of changes in and on the graph of

1) interpret roots to a quadratic equation in context of parabolic motion 2) find roots to quadratics using calculators 3) find roots to quadratics by factoring 4) sketch a graph of quadratic equations

1) interpret roots to a quadratic equation in context of parabolic motion 2) find roots to quadratics using calculators 3) find solutions to quadratics by taking square roots

1) use completing the square to find the roots of a quadratic equation 2) derive and use the quadratic equation 3) interpret solutions in the context of a meaningful situation 4) find solutions to quadratics by taking square roots

1) use completing the square to derive the quadratic formula 2) use the quadratic formula to find roots 3) use the discriminant to determine the number of roots to a quadratic

P: Project kickoff; anchor video

How can we find where the football lands?

Is there an alternative to factoring?

Is there an alternative to factoring?

Using completing the square, can we find a formula for the roots?

LA: Students will learn that parabolic motion is modeled by a quadratic equation. Lecture over how to find key points on the graph and learn to find the y-intercept and vertex algebraically. Practice problems. Lecture and practice problems over how to modify the parent function to shift, scale, and change orientation

Warm-up: sketching quadratics Notes over solving by factoring, solving by calculators From solving by factoring, students will infer how to find a quadratic equation with given roots. Project work time

Warm-up: solving by factoring Students will learn how to solve quadratics using square roots, and use the square root function on the calculator for large numbers. Project work time

Warm-up: solving by square roots. Working in groups, students will infer the formula to make a perfect square trinomial. Notes over completing the square. Project work time

Students will present HW question about completing the square on Students will learn the quadratic formula song. Elaboration: using the determinant to identify the number of roots

N: HW: graphing quadratics, shifting quadratics

HW: solving quadratics, graphing and shifting quadratics

Assessment: Quiz over graphing quadratics HW: solving quadratics, graphing and shifting quadratics

HW: completing the square

Assessment: Quiz over solving quadratics Homework: quadratic formula practice

Group members ________________________________________________________________

Robot Quarterback Project Outline

You have been asked by your high school to help the football team succeed in the coming up season. We recently saw on FOX news that the Dallas Cowboys are using a robot quarterback to help train their receivers. We want to replicate this robot quarterback for our own team. This robot is going to have to train several different players and use many plays. We can’t have just one play programmed on the robot because that will make our team too predictable. We are asking you to help program the robot. We will review the plays you have come up with at the end of the two week period you have to complete this project. Your final goal should be to have your play picked by the football team in order to program the school’s new robot quarterback. Good Luck! Driving question: How should the quarterback throw the ball to get it over defenders and to the receiver? There will be one smaller project assignment (counting for 2 daily grades) to prepare you for the final product (1 test grade). The entire project is posted at algebraandcoachjrock.weebly.com. The final submission will be an equation for the path of the ball which satisfies your football play (the football must travel from the quarterback to the receiver and above any defenders). More details are on the back. Lessons will cover material necessary to complete the project. Class time will be given after completing the notes. Quizzes and tests will cover material on the homework, which will be reinforced by the project. Deadlines: Wednesday 4/3 1. Football play (part of final product) Friday 4/5 2. Where will the football land? (2 daily grades) Thursday 4/11 3. Completions and interceptions (part of final product) Wednesday 4/17 Final product (1 test grade) Monday 4/3 Quiz 1 (graphing) (1 daily grade) Tuesday 4/9 Quiz 2 (mixed factoring and solving quadratics) (1 daily grade) Monday 4/15 Test (1 test grade) Late penalty: -10% per school day late. Assignments turned in after the end of the period will be counted as 1 day late.

Presentation: The final submission should be neatly handwritten or typed and presented on poster board.

Robot Quarterback Final Product (due Wednesday 4/17 at start of class) o A drawing or image of the football play, with robot quarterback, receivers, and defenders

labeled (assignment 1) (5%)

o A number line with labeled positions of the robot quarterback, receivers, and defenders in the football play with proper units (assignment 1) (5%)

o An equation describing a path for the football which takes it from the robot quarterback,

over all defenders, and to the receiver. (assignment 3) (25%) 1. The robot quarterback throws the football from 4ft 2. The receiver catches the football at 4ft. 3. Defenders can block a football between ground level and 7ft.

o Using the equation of the football, verify the above 3 parts algebraically. (assignment 3)

(15%) o From the equation in assignment 3, determine where the football will hit the ground if it is

not caught by the receiver. Use the quadratic formula. Give the answer with proper units. There should be 1 point for where the football will hit the ground if it is not caught

by the receiver. Do not give 2 points as the final answer. (10%)

o Sketch the path of the football from assignment 3, and label the following: (30%)

1. Axes 2. The ground 3. The positions of the robot quarterback, receivers, and defenders 4. The maximum height reached by the football 5. Where the football will hit the ground if it is not caught by the receiver

Format/presentation (10%)

Rubric

Football play 0 1-4 5

Drawing/image of play

Graphic is missing or impossible to read

Graphic shows players, but missing labels or difficult to follow

Graphic shows labeled positions of players

Number line Number line is missing or impossible to read

Points on the line are not labeled, or positions do not correspond to the football play graphic

Labeled points on the line indicate positions of the quarterback, receiver, and defenders that match the play graphic

Equation 0 1-9 10

Quadratic The equation is not for a parabola.

The equation is for a parabola which could not model the path of the football.

The equation is for a parabola which could model the path of the football based on the vertex and orientation

Robot quarterback

The location of the robot quarterback is not on the path of the football.

The location of the robot quarterback is on the path of the football, but is not checked algebraically.

The location of the robot quarterback is on the path of the football, and is algebraically verified.

Receivers The football does not reach the receiver.

The football reaches the receiver, but it is not checked algebraically.

The football reaches the receiver between 3ft and 6ft, and the height of the football at the receiver’s position is verified algebraically.

Defenders The football is blocked by a defender.

The football passes above all defenders, but it is not checked algebraically.

The football passes above all defenders, and the height of the football at the receiver’s position is verified algebraically.

Where the football will hit the ground

0 1-9 10

Answer is missing or randomly chosen.

Few algebraic errors, or the appropriate solution is not given

The appropriate root of the parabola is found algebraically and given as the answer.

Graph 0 1-19 20

Graph Graph is missing or mostly incorrect.

The graph is a partially correct image of the

The graph of the parabola correctly shows the

equation, or extends past the meaningful domain.

equation in the meaningful domain.

Labels The following are labeled on the graph (10pts) 1. 𝑥-axis 2. 𝑦-axis 3. the ground 4. robot quarterback 5. receiver 6. 1st defender 7. 2nd defender 8. Vertex 9. where the football will hit the ground if it is not caught by the receiver

Format / presentation

0 1-9 10

Not presented on poster board

Format on poster board is difficult to follow, or project is difficult to read

Neatly written or typed on poster board (or on printer/construction paper attached to poster board)

Robot Quarterback assignment 1: Football play (due at end of class on Wednesday 4/3)

(part of final product) Write neatly on printer or construction paper. This will be attached to the poster as part of the final product.

a) Choose a football play to program into the robot quarterback. The play must include the quarterback throwing the football to the receiver, with at least 1 defender trying to block the quarterback, and at least 1 defender guarding the receiver (at least 2 defenders total) b) Draw or find an image of the positions of the quarterback, receiver, and defenders on the field during the football play you chose in part a). Label the players.

c) Use the markings on the field in the image in part b) to estimate the scale of the image. i.e. 1in on the image = _____ft d) Based on the image in part b), and the scale in part c), estimate the positions of the quarterback, receiver, and defenders on a number line. Label those players. Use proper units.

Robot Quarterback assignment 2: Where will the football land? (due at end of class on Friday 4/5)

(2 daily grades)

a) Suppose the robot quarterback launches the football from ground level at (5,0). The football lands on the ground at (−3, 0). Find the equation of a parabola through those points. b) Modify the equation from part a) so that the orientation of the parabola resembles the path of a football (opens down). c) Find the vertex of the parabola for the equation in part b). What is the maximum height reached by the football for the equation in part b)? d) Multiply the entire equation in part b) by a constant so that the maximum height reached by the football is 8ft.

o Check that the path of the football still passes through the robot quarterback at (5,0) and the ground at (−3, 0). (Use a calculator).

e) Put the quadratic in part d) in standard form.

f) Sketch a graph of the equation in part d)

g) On the graph, label the following: o Axes o The position of the Robot Quarterback o The ground o The maximum height reached by the football o Where the football hits the ground o Where the football crosses the 𝑦-axis

Robot Quarterback assignment 3: Completions and interceptions (due at start of class on Thursday 4/11)

(part of final product)

Write neatly on printer or construction paper. This will be attached to the poster as part of the final product.

For greater realism in simulating a human quarterback, the robot quarterback has been elevated to 4ft. The human receiver is practicing catching at 4ft also. a) Using the number line from assignment 1, find an equation of a parabola that goes from the robot quarterback at ground level to the human receiver at ground level. b) Modify the equation from part a) so that the parabola resembles the path of a football (opens down). c) Modify the equation from part c) so that the robot quarterback throws the football from 4ft and the human receiver catches the football at 4ft.

d) Stretch the parabola in part c) so that the football passes above all defenders. Defenders can block the football from ground level to 7ft. e) Using the equation of the parabola from part c), verify algebraically that

1. the football originates from the robot quarterback at 4ft 2. the football is at 4ft when it passes the 𝑥-coordinate of the receiver 3. the football is above 7ft when it passes the 𝑥-coordinates of all defenders. (If any of the above are false, you will need to fix your equation)

Robot Quarterback Final Product (due at start of class Wednesday 4/17)

Presentation: The final submission should be neatly handwritten or typed and presented on poster board.

o A drawing or image of the football play, with robot quarterback, receivers, and defenders

labeled (assignment 1)

o A number line with labeled positions of the robot quarterback, receivers, and defenders in the football play with proper units (assignment 1)

o An equation describing a path for the football which takes it from the robot quarterback,

over all defenders, and to the receiver. (assignment 3) 1. The robot quarterback throws the football from 4ft 2. The receiver catches the football at 4ft. 3. Defenders can block a football between ground level and 7ft.

o Using the equation of the football, verify the above 3 parts algebraically. (assignment 3) a) From the quadratic in assignment 3, find the maximum height reached by the football.

b) From the equation in assignment 3, determine where the football will hit the ground if it is

not caught by the receiver. Use the quadratic formula. Give the answer with proper units. There should be 1 point for where the football will hit the ground if it is not caught

by the receiver. Do not give 2 points as the final answer. c) Sketch the path of the football from assignment 3, and label the following with proper units:

1. Axes 2. The ground 3. The positions of the robot quarterback, receivers, and defenders 4. The maximum height reached by the football 5. Where the football will hit the ground if it is not caught by the receiver

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Name __________________________ Robot Quarterback Day 1 NOTES – Pre-AP

Sketch the path of a football thrown from a quarterback to a receiver.

We call this shape a ________________. It is described by a ____________________ equation, which is a polynomial of degree _______. What is the parent function? _________________ Graph it on your calculator. Make a table of points, then sketch the graph: 𝑥 𝑦 How many 𝑥-intercepts does it have? _______ How many 𝑦-intercepts does it have? _______ How do you find the 𝑦-intercept? ________________________________________________ What are other names for 𝑥-intercepts? _____________________________________________

Does it open up or down? __________ Domain: __________________________ Range: ___________________________ Which value of 𝑥 results in the minimum value of 𝑦? ______________________ We call this point the ____________________. For 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐, the 𝑥-coordinate is 𝑥 =____________. Memorize this! Also, remember the vertex is a ___________. The axis of symmetry is the line 𝑥 =____________. It passes through the _____________.

quarterback receiver

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Practice: Find the vertex and 𝑦-intercept.

1. 𝑦 = 𝑥2 + 2𝑥 + 1 2. 𝑦 = 𝑥2 − 9 3. 𝑥2 + 4𝑥 + 10

𝑎 = __________________ __________________ __________________ 𝑏 = __________________ __________________ __________________ vertex: __________________ __________________ __________________ axis of __________________ __________________ __________________ symmetry

𝑦-intercept: __________________ __________________ __________________

That is flipped compared to the path of a thrown football. Change the equation to get a graph with the correct orientation.

Equation: ___________________________________

Make a table of points and sketch the graph: 𝑥 𝑦 + 3

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Domain: _________________ Range: _______________ Does it open up or down? Modify the equation 𝑦 = 𝑥2 to shift it up by 3: ______________________________

Give the equation of a parabola shifted down by 2: ______________________________

Modify the equation 𝑦 = 𝑥2 to shift it down by 4 and have it open down :

______________________________

Modify the equation 𝑦 = 𝑥2 to vertically stretch it by a factor of 2.

Equation: ___________________________________

Make a table of points and sketch the graph: 𝑥 𝑦 Modify the equation 𝑦 = 𝑥2 to vertically stretch it by 3: ______________________________ Modify the equation 𝑦 = 𝑥2 to vertically stretch it by 10: ______________________________

Modify the equation 𝑦 = 𝑥2 to vertically stretch it by 8 and have it open down :

______________________________

Modify the equation 𝑦 = 𝑥2 to horizontally stretch (vertically shrink) it by a factor of 2: ______________________________

Modify the equation 𝑦 = 𝑥2 to horizontally stretch it by 10: _____________________________ Modify the equation 𝑦 = 𝑥2 to vertically shrink it by 5: _____________________________ Modify the equation 𝑦 = 𝑥2 to horizontally stretch it by 2 and shift it up by 3: _____________________________

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Name ________________________ Robot Quarterback Day 1 HW – Pre-AP

Answer the following questions about each parabola and sketch the graph.

1. 𝑦 = 3𝑥2 − 3𝑥 a) opens up / down b) 𝑦-intercept: _______ 𝑥 𝑦 c) domain: _________ d) range: _________ e) vertex: __________ f) # of 𝑥-intercepts: ______ g) axis of symmetry: _____________ 2. 𝑦 = 3 − 2𝑥 − 𝑥2 a) opens up / down b) 𝑦-intercept: _______ 𝑥 𝑦 c) domain: _________ d) range: _________ e) vertex: __________ f) # of 𝑥-intercepts: ______ g) axis of symmetry: _____________ 3. 𝑦 = 𝑥2 + 4𝑥 + 4 a) opens up / down b) 𝑦-intercept: _______

c) domain: _________ d) range: _________ e) vertex: __________ f) # of 𝑥-intercepts: ______ g) axis of symmetry: _____________

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4. Find a quadratic with 0 𝑥-intercepts. Give the equation: ____________________________ Sketch the graph. Modify the parent function 𝑦 = 𝑥2 to produce the following transformations, and sketch the graph. 5. Shifted up by 5. 6. Opens down and shifted up by 3. ______________________________ ______________________________ 7. Vertically stretched by a factor of 4 8. Opens down and horizontally

stretched by a factor of 3 ______________________________ ______________________________

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Name __________________________ Robot Quarterback Day 2 NOTES – Pre-AP

Warm-up: 1. How do you find the vertex of ?

2. Suppose the path of the football is described by . Sketch a graph and answer the following.

a) # of -intercepts: ______ b) domain: _________ c) range: _________ d) opens up / down e) y-intercept: _________ f) vertex: ___________ g) axis of symmetry: ___________ Standard form for a quadratic is similar to __________________________________________

Put the following quadratics in standard form.

3. 4. 5. Suppose the path of a football is described by . Give an equation that describes the -coordinate(s) of when the football is at ground level.

____________________________________________ How can we solve this equation? ________________________________

Find the solutions.

Practice: algebraically find the roots to the following equations.

6. 7. 8.

Finding roots using a graphing calculator.

Step 1: graph the equation ( ) Step 2: Press 2nd CALC. Select zero.

Step 3: move the cursor to the left of one root. Press ENTER. Move the cursor to the right of

one root. Press ENTER again. Press ENTER once more.

Step 4: Repeat step 3 for the second root.

Practice: Use the calculator to find solutions to the following:

9. ____ , ____ 10. ____ , ____

11. Fill in the blanks so the solutions are . Check by solving the equation. . 12. Fill in the blanks so the solutions are . Check by solving the equation. . 13. Fill in the blanks so the solutions are : .

14. Find a quadratic with roots : .

15. Find a quadratic with roots : ______________________________

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Name __________________________ Robot Quarterback Day 2 HW – Pre-AP

Sketch a graph and answer the following questions about each quadratic.

1. a) opens up / down b) -intercept: _______ c) domain: _________ d) range: _________ e) vertex: __________ f)axis of symmetry: ___________ g) Find the -intercepts by factoring. 2. a) opens up / down b) -intercept: _______ c) domain: _________ d) range: _________ e) vertex: __________ f)axis of symmetry: ___________ g) Find the -intercepts by factoring.

Put the following quadratics in standard form.

3. 4. 5.

Find roots by factoring.

6. 7.

8. 9. 10.

Modify the parent function to produce the following transformations.

11. Shifted up by . 12. Opens down and vertically 13. Horizontally stretched by 10 and stretched by . shifted down by 7.

Find the equation of a quadratic with the given roots.

14. 15. 16.

Name __________________________ Robot Quarterback Day 3 NOTES – Pre-AP

Warm-up: Find all solutions by factoring. 1. 𝑦 = 𝑥2 − 7𝑥 + 10 2. 𝑥2 = 4 3. 𝑥2 = 16

4. Solve using square roots: √𝑥2 = √4 5. Solve using square roots:√𝑥2 = √16 Check with warm-up #2: _____ Check with warm-up #3: _____ When solving using square roots, remember ____________________________________

Solve using square roots.

6. 𝑦2 = 81 7. 𝑧2 = 25 8. 2𝑥2 = 2 Find solutions using square roots.

9. (𝑥 − 7)2 = 16 10. 𝑦 = (𝑥 + 1)2 − 4 11. 2(𝑥 + 2)2 = 18 12. 𝑦 = 3(𝑥 − 3)2 − 75

Use the calculator to solve the following.

13. 3𝑥2 = 49 14. 𝑦2 = 9025 15. 3𝑧2 = 8112

Perfect squares cheat sheet

12 = 1

22 = 4

32 = 9

42 = 16

52 = 25

62 = 36

72 = 49

82 = 64

92 = 81

102 = 100

112 = 121

122 = 144

132 = 169

142 = 196

152 = 225

162 = 256

172 = 289

182 = 324

192 = 361

202 = 400

212 = 441

222 = 484

232 = 529

242 = 576

252 = 625

262 = 676

272 = 729

282 = 784

292 = 841

302 = 900

312 = 961

322 = 1024

332 = 1089

342 = 1156

352 = 1225

362 = 1296

372 = 1369

382 = 1444

392 = 1521

402 = 1600

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Name ________________________ Robot Quarterback Day 3 HW – Pre-AP

Sketch the graph and answer the questions about the parabola. 1. 𝑦 = 25 − 𝑥2 a) # of 𝑥-intercepts: ______ b) range: _________ c) opens up / down (circle one) d) 𝑦-intercept: _________ e) vertex: ___________ f) axis of symmetry: _________ g) Find the 𝑥-intercepts using square roots. Find solutions using square roots.

2. 𝑥2 = 49 3. 𝑦2 = 196 4. 𝑦 = 49𝑥2 − 625 5. 𝑦 = 4𝑥2 − 121 Find solutions using square roots.

6. (𝑥 + 3)2 = 144 7. 𝑦 = (𝑥 − 2)2 − 81 8. 𝑦 = 4(𝑥 − 10)2 − 400

Find roots to the following. Use any method. (Some require factoring).

9. 𝑦 = 4𝑥 − 2𝑥2 10. 𝑦 = (𝑥 − 3)2 − 25

11. 𝑦 = 𝑥2 + 4𝑥 − 21 12. 𝑦 = 4𝑥2 − 49

13. 𝑦 = 5(𝑥 − 1)2 − 125 14. 𝑦 = 2𝑥2 − 5𝑥 + 3

Name __________________________ Robot Quarterback Day 4 NOTES – Pre-AP

Sometimes factoring is difficult. We will learn another way to factor 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0.

Warm-up: solve by square roots.

1. (𝑥 − 7)2 = 16 2. (𝑥 + 1)2 = 64 3. (𝑥 + 2)2 = 121 The challenge today is putting the quadratic in the above form. We call this ___________________________________________________. Recall that (𝑎 + 𝑏)2 =___________________________.

For example, 𝑥2 + 2𝑥 + 1 = (𝑥 + 1)2 is a perfect square trinomial.

Practice: What constant makes a perfect square trinomial? 1. 𝑥2 + 4𝑥 + _______ 2. 𝑥2 + 6𝑥 + _______ 3. 𝑥2 − 2𝑥 + _________ 4. 𝑥2 − 8𝑥 + _________ Write 𝑥2 + 4𝑥 + 3 = 0 with a perfect square trinomial in the parentheses. (𝑥2 + 4𝑥 + ___________) + 3 = 0 + ___________ Hint: add to both sides

Now factor the trinomial in parentheses (it should be a perfect square). (𝑥 + _________)2 + 3 = 0 + __________ Solve using square roots. Write 𝑥2 + 8𝑥 + 12 = 0 with a perfect square trinomial in the parentheses. (𝑥2 + 8𝑥 + ___________) + 12 = 0 + ___________ Hint: add to both sides

Now factor the trinomial in parentheses (it should be a perfect square trinomial). (𝑥 + _________)2 + 12 = 0 + __________ Solve using square roots.

Practice: write 𝑥2 + 2𝑥 − 15 = 0 with a perfect square trinomial in the parentheses: (𝑥2 + 2𝑥 + ___________) − 15 = 0 + ___________ Hint: add to both sides

Now factor the trinomial in parentheses (it should be a perfect square trinomial). (𝑥 + _________)2 − 15 = 0 + __________ Solve using square roots. Find solutions by completing the square.

1. 𝑥2 + 8𝑥 + 12 = 0 2. 𝑥2 + 4𝑥 − 12 = 0 3. 𝑦 = 𝑥2 − 10𝑥 + 16 4. 𝑦 = 𝑥2 + 6𝑥 + 8 5. 2𝑥2 − 8𝑥 = 10 6. 𝑦 = 3𝑥2 − 12𝑥 − 36

Generalize: for 𝑥2 + 𝑏𝑥 + 𝑐 = 0, rewrite as (𝑥 + ______)2 + _______ = _______ . Then ____________________________________________________________________. For 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0, first __________________________________________________.

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Name __________________________ Robot Quarterback Day 3 HW – Pre-AP

Solve using square roots.

1. 2. 3.

Fill in the blank to make a perfect square trinomial.

4. ______ 5. ______ 6. ______ 7. ______

Find all solutions by completing the square.

8. 9.

10. 11.

12. a) Find the equation of a quadratic that has solutions at and opens down.

b) Graph the quadratic. c) y-intercept: ______ d) vertex: __________ e) axis of symmetry: _______ f) domain: ______ g) range: ______ h) maximum or minimum value: _____

Find all solutions by completing the square.

13. 14.

15. 16.

17. Use completing the square to solve . Show all work. Your answer should be in terms of , , and . We will need this answer for Friday.

Quadratic formula song Quadratic formula

equals negative , For ,

Plus or minus the square root,

Of squared minus ,

All over .

𝑥 −𝑏 ± 𝑏 − 𝑎𝑐

𝑎

Name __________________________ Robot Quarterback Day 4 NOTES – Pre-AP

Warm-up: Find solutions by completing the square.

1. 𝑥2 + 10𝑥 + 16 = 0 2. 𝑦 = 𝑥2 − 2𝑥 − 24

Practice: find all solutions.

3. 𝑥2 + 6𝑥 + 5 = 0 4. 2𝑥2 + 9𝑥 + 4 = 0 5. 𝑦 = 𝑥2 − 9𝑥 + 20 6. 𝑦 = 𝑥2 − 3𝑥 − 18

For 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0, 𝑥 =

This is called the ________________________________________________________

Goal for today: a formula that gives solutions to any quadratic.

What happens if you have a square root of a negative? Solve 𝑥2 + 2𝑥 + 10 = 0 using the quadratic formula. Graph on the calculator. How many solutions does it have? _________

Practice: find all solutions using the quadratic formula.

7. −4𝑥2 + 3𝑥 − 1 = 0 8. 2𝑥2 + 3𝑥 + 5 = 0 9. 𝑦 = 𝑥2 + 10𝑥 + 25 10. 𝑦 = 4𝑥2 + 2𝑥 + 3

We call _____________________________ the ___________________________________. If it is zero, then the quadratic has _____ real number solutions. If it is negative, then the quadratic has _____ real number solutions.

Name __________________________ Robot Quarterback Day 5 HW – Pre-AP

1. What is the Quadratic Formula for solving ?

Use the discriminant to find the number of real solutions to each quadratic. If there is at least

1 real number solution, use the quadratic formula to find all solutions.

2. 3.

4. 5.

6. 7.

Use the discriminant to find the number of real solutions to each quadratic. If there is at least

1 real number solution, use the quadratic formula to find all solutions.

8. 9.

10. 11.