Algebra 1 Unit 9 Quadratic Equations Part 1 Name: Period: Date Name of Lesson Notes Tuesda 4/4 Wednesda 4/5 Thursda 4/6 Frida 4/7 Monda 4/10 Tuesda 4/11 Wednesda 4/12 Thursda 4/13 Frida 4/14 Da 1- Quadratic Transformations Da 2- Verte Form of Quadratics Da 3- Solving Quadratics Using Graphs Da 4- Appling Graphs Activit Da 5- Solving Quadratics Using Square Roots Da 6- Solving Quadratics Review Da 7- Quadratics Quiz Da 8- Verte Form of Quadratics, Part 1 Da 9- Verte Form of Quadratics, Part 2
Quadratic Transformations Guided Notes Da 1 Warm-up: Graph the following Absolute Value Functions and give the domain and range 1. = 3 4 2. f() = 3 1 + 5 Lesson Objective: Notes Quadratic Function: The parent functions is: -2-1 0 1 2 Graph: = + 2 4 Graph: : = ( + 2) 2 4 Domain: Range: Describe the transformations -4-3 -2-1 0 Domain: Range: Describe the transformations What do ou notice? Graph: = 2 3 Graph: = 2 2 3 Domain: Range: Describe the transformations -2-1 0 1 2 Domain: Range: Describe the transformations Describe how ou can determine the domain and range for both functions.
Graph: = 1/2 1 + 3 Graph: = 1/2( 1) 2 + 3 Domain: Domain: -1 Range: 0 Range: 1 Describe the 2 Describe the transformations 3 transformations 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: a: + a: a > 1 a < 1 = a( h) 2 + k h: k: Practice Problems Graph and give the domain and range for each of the quadratic functions = 2 3 f() = ( + 2) 2 + 3 g() = 2( 2) 2 5 Translate the graph of f()= 2 seven units to the left, si units down and verticall stretch the graph b a factor of 12. What is the new equation? Below is a graph for the p profits for varous selling prices of a skateboard. At what selling price should ou choose to make a maimum profit? At what selling prices would give ou a profit of $6000? At what selling prices would ou make a $0 profit?
Algebra 1 Da 2 Verte Form of Quadratic Functions Name Date Period Toda s Lesson Goals: Understand and identif the vocabular/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() = ( 3) 2 4. The name for the graph of quadratic function is a. Label the parts of a graph of a quadratic function: Verte Line/Ais of Smmetr Real Zeros -intercept Tr these! f() = ( + 3) 2 1 f() = 2( 3) 2 + 8 Predict Verte: Predict Verte: Ais of Smmetr: Real Zeros: -intercept: Verif Graphicall Ais of Smmetr: Real Zeros: -intercept: Verif Graphicall Think about it!
1. What relationships do ou see between the following: Verte and the Line of Smmetr Verte and the Real Zeros Line of Smmetr and the Real Zeros Part 2: Graph the following quadratic functions f() = 1 2 ( + f() = ( + 1) 1)2 2 2 f() = ( 2) 2 4 # of real zeros: # of real zeros: # of real zeros: Think about it! 1. How can ou predict the number of real zeros from the equation?
Solving Quadratic Equations Graphicall da 3 Notes Warm-up: Graph: Graph: Graph: = 2 4 = 2 + 3 = - 4 Solve the absolute value equation graphicall 1/2 + 2 3 = + 1 f() = How man solutions do ou think ou can have when ou solve an equation with an absolute value function and a linear function? Describe how ou know this? g() = *Which coordinate are we wanting to find? Objective: Notes: Determine the number of real solutions for the following equations. 1. Graph to determine the How man solutions does this equation have? number of solutions f() = ( 2) 2 + 1 = 3 Even though ou ma not be able to determine the eact solution, what would be the approimate solution(s)? g() = 2. Graph to determine the number of solutions How man solutions does this equation have? f() = 2 2 + 1 = 1 2 + 1 Even though ou ma not be able to determine the eact solution, what would be the approimate solution(s)? g() = 3. Graph to determine the number of solutions ( 2) 2 + 2 = ( 2) 2 + 3 f() = How man solutions does this equation have? Even though ou ma not be able to determine the eact solution, what would be the approimate solution(s)? g() =
Determine the solutions for the following equations. 1. Graph to solve the 2. Graph to solve the equation. equation. 2 3 = 1 ( + 2) 2 + 1 = 3 f() = g() = What is the solution(s)? f() = g() = What is the solution(s)? 3. Graph to solve the equation. ( + 1) 2 + 1 = + 2 f() = g() = What is the solution(s)? 4. Graph to solve the equation. f() = g() = ( 3) 2 = ( 1) 2 What is the solution(s)? Summarize: How do ou solve an equation using graphs? What are the pros for using this method? What cons for using this method? (share with our partner when ou are done)
Algebra 1 Da 4 Notes Angr Birds Applications Name 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 -intercept represent? 3. What do the -intercepts represent? 4. The highest part of the bird s flight is represented b what part of the parabola? 5. Answer the following: a. How far does Angr Bird fl in h()? b. How high does he go? c. How far awa from the catapult is he when he is at his highest? d. When he is 15 feet awa, how high is he fling?
Round 2: Using the line of smmetr 1. When Angr Bird is 9 feet awa, how high is he fling? 2. The ais of smmetr is provided. What part of the parabola does this pass through? What does this part represent about Angr Bird s flight? 3. How high does the bird fl? 4. Reflect points over the ais of smmetr to complete the parabola. Do ou hit an pigs? 5. How far would Angr Bird fl if he did not hit an obstacles? 6. Without solving for the whole equation: Is a positive or negative? Round 3: Using the Quadratic Function 1. Angr bird wants to hit the hungr pig on the left. Angr bird and hungr pig are 18 feet awa from each other and are at same height (-value) when angr bird is catapulted. At what distance awa will Angr Bird be the highest? Think about smmetr. 2. Now Angr Bird wants to hit the pig on the right. The equation representing his flight is: = 0.083( 10.964) 2 +9.977 a) Using the picture, what is the -intercept? b) Using the picture, what are the -intercepts? c) Where is the ais of smmetr? d) How high does Angr Bird fl (rounded to the nearest integer)? e) Sketch the graph of Angr Bird s flight.
Time Permitting: Write the equation for the quadratic ou sketched. Use DESMOS to verif that it works. If not, keep re-writing an equation until ou found the perfect (or almost perfect one).
Lesson Objective: Solving Quadratic Equations Using Square Roots da 5 Notes Isolate the variable or epression being squared (get it ) Square root both sides of the equation (include + and on the right side!) This means ou have equations to solve!! Solve for the variable (make sure there are no square roots in the denominator) Simplif our square roots 1) 3 2 = 75 2.) 3 2 12 = 69 3.) 4 2 1 = 0 4.) 2 m 15 3 2 5.) ( + 3) 2 = 49 6.) ( - 2) 2-6 = 34 7) 1 2 ( 1)2 8 = 0 8.) 3(a + 5) 2 + 24 = 0 Can ou think of another wa ou could solve these equations? How solving square root equations be useful for tomorrow s objective?
Verte Form of Quadratic Functions Algebraicall Da 8 and 9 Notes Warm-up: A soccer ball is kicked from ground level with an upward velocit 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 maimum height the ball goes? Which part of the equation tells ou this? b. How long does it take for the ball to reach its maimum height? Which part of the equation tells ou this? c. How long is the ball in the air? Justif our answer. d. What is the height of the ball after 1 second? Show the work that gives ou this answer. e. At what times is the ball 100 feet above the ground? Show the work that gives ou this answer. Lesson Objective: Notes: Determine the verte, zeros, and -intercept from verte form of a quadratic function. Eample 1 How do ou find the Zeros? How do ou find the - Intercept? = 2 25 Verte: Minimum or Maimum What is the value? Eample 2 How do ou find the Zeros? How do ou find the - Intercept? = ( + 2) 2 + 45 Verte: Minimum or Maimum What is the value? Wh would finding a verte be important in a real world eample? Wh would finding the zeros be important in a real world eample? Wh would finding a -intercept be important in a real world eample?
Appling verte form to real world situations Christopher is repairing the roof on a shed. He accidentl dropped a bo of nails from a height of 14 feet.. This is represented b 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 bo of nails to hit the ground? (Do we need to find the verte or the zeros? The opening of a one lane tunnel shown in the graph can be modeled b the quadratic equation. (look familiar ) = 0.18( 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 verte, zeros, and -intercept of the equation: f() = ( + 3) 2 9 Verte: Zeros: Determine the verte, zeros, and -intercept of the equation: f() = 3( 1) 2 15 Verte: Zeros: -intercept: -intercept Della s parents are throwing a Sweet 16 Part 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?