Unit 2: Quadratic Functions and Modeling. Lesson 3: Graphing Quadratics. Learning Targets: Important Quadratic Functions Key Terms.

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1 Unit 2: Quadratic Functions and Modeling Lesson 3: Graphing Quadratics Learning Targets: - Students can identify the axis of symmetry of a function. - Students can find the vertex of a quadratic - Students can graph a quadratic. - Students can find the min/max (vertex), zeroes, and y-intercept of a quadratic - Students can use properties of quadratics to solve real-life problems. Important Quadratic Functions Key Terms Vertex: Axis of Symmetry: X-intercepts (Zeroes/Roots): Y-intercepts: Identify the vertex, zeroes, axis of symmetry and y-intercept of the following graph.

2 Try These

3 Investigation #13: Pumpkins in Flight Punkin Droppin At Old Dominion University in Norfolk, Virginia, physics students have their own flying pumpkin contest. Each year they see who can drop pumpkins on a target from 10 stories up in a tall while listening to music by the group Smashing Pumpkins. By timing the flight of the falling pumpkins, the students can test scientific discoveries made by Galileo Galilei, nearly 400 years ago. Galileo used clever experiments to discover that gravity exerts force on any free-falling object so that d, the distance fallen, will be related to time t by the function d = 16t 2 (time in seconds and distance in feet) For example, suppose that the students dropped a pumpkin from a point that is 100 feet above the ground. At a time 0.7 seconds after being dropped, the pumpkin will have fallen 16(0.7) feet, leaving it = feet above the ground. This model ignores the resisting effects of the air as the pumpkin falls. But, for fairly compact and heavy objects, the function d = 16t 2 motion of falling bodies quite well. Fill in the table below to show estimates for a pumpkin s distance fallen and height above ground in feet at the various times given between 0 and 3 seconds. Time t Distance Fallen d Height Above Ground h = Use the data relating HEIGHT and TIME to answer the following questions about flight of a pumpkin dropped from a position 100 feet above the ground. a. The function rule that shows how the pumpkin s height h(t) is related to time t is h(t) =. b. Estimate the time when the pumpkin is 10 feet from the ground. (Hint: Try graphing the function from part (a) and use window 0 x 4, scl 0.25, and -10 y 100, scl 10) c. (Estimate) when the pumpkin will hit the ground.

4 d. If the pumpkin were to be dropped from a spot 75 ft above the ground i. What is the new function rule relating pumpkin s height h(t) and time t? ii. Estimate the time when the pumpkin is 10 feet from the ground. iii. When will the pumpkin hit the ground? Give your best estimate for the solution. High Punkin Chunkin Compressed-air cannons, medieval catapults,and whirling slings are used for the punkin chunkin competitions. Imagine pointing a punkin chunkin cannon straight upward. The pumpkin height at any time t will depend on its speed and height when it leaves the cannon. 2. Suppose a pumpkin is fired straight upward from the barrel of a compressed-air cannon at a point 20 feet above the ground, at a speed of 90 feet per second (about 60 miles per hour!). a. If there were no gravitational force pulling the pumpkin back toward the ground (and no wind, air resistance, etc), how would the pumpkin s height above the ground change as time passes? b. The new function rule relating the height above the ground h(t) in feet to time in the air t in seconds is h(t) =. c. If the punkin chunker used a stronger cannon that fired the pumpkin straight up into the air with a velocity of 120 feet per second, the function rule in Part b changes to h(t) =. d. If the end of the cannon barrel was only 15 feet above the ground, instead of 20 feet, the new function would be h(t) =. 3. a) Suppose a compressed-air cannon fires a pumpkin straight up into the air from a height of 20 feet and provides an initial upward velocity of 90 feet per second. Write a function rule that gives the pumpkin s height h(t) after t seconds make sure you include the initial height of the pumpkin s release, the initial upward velocity, and gravity. h(t) = b) A function rule relating the pumpkin s height h(t) and its time t of a pumpkin launched at a height of 15 feet with an initial upward velocity of 120 feet per second is given by h(t) =

5 4. By now you may have recognized that the height of a pumpkin (shot straight up into the air) at any time in its flight will be given by a function that can be expressed with a rule in the general form h(t) = h 0 + v 0 t 16t 2 a. What does the value of h 0 represent? What units are used to measure h 0? b. What does the value of v 0 represent? What units are used to measure v 0? 5. The pumpkin s height in feet t seconds after it is launched will still be given by h(t) = h 0 + v 0 t 16t 2. It is fairly easy to measure the initial height (h 0 ) from which the pumpkin is launched, but not so easy to measure the initial upward velocity (v 0 ). a. Suppose that a pumpkin leaves a cannon at a point 24 feet above the ground when t = 0. Substitute this value into the appropriate spot in the height function. b. Suppose you were able to use a stopwatch to discover that the pumpkin shot described above returned to the ground after 6 seconds. Use that information to find the value of v 0 (don t forget your units!).

6 Investigation 13: Pumpkins In Flight In this investigation, you used several strategies to find rules for quadratic functions that relate the position of flying objects to time in flight. You used those function rules and resulting tables and graphs to answer questions about the problem situations. a. How can the height from which an object is dropped or launched be seen: i. In a table of (time, height) values? ii. On a graph of height over time? iii. In a rule of the form h=h 0 +v 0 t-16t 2 b. How could you determine the initial upward velocity of a flying object from a rule in the form h=h 0 +v 0 t-16t 2 giving height as a function of time? c. What strategies can you use to answer questions about the height of a flying object over time? Be prepared to share your ideas and strategies with others in your class.

7 Investigation 14: Patterns in Tables, Graphs and Rules Expressions in the general form ax 2 + b x + c are called quadratic functions. In this investigation, you are going to study the patterns that develop in the tables and graphs of quadratic functions, and how the values of a, b, and c effect quadratic functions. The Basic Quadratic Function When distance traveled by a falling pumpkin is measured in feet, the rule as a function of time is d = 16t 2. When such gravitational effects are studied on the Moon, the rule becomes 2 d = 2.6t. When distance is measured in meters, the rule is d = on Earth and d = 0.8t. These are all examples of the simplest quadratic functions those defined by rules in the form y = ax 2. How can you predict the shape and location of graphs of quadratic functions with rules in the form y = ax 2? Graph the quadratic function y = x 2 using the standard window What shape is the graph? Is the graph symmetrical? 2. Describe the rate of change in detail below. 3. At what point on the parabola does the rate of change switch? 4. Look at the table for the same function [TblStart = -10]. a. Copy the table below. x Y b. Do you see the same rate of change in the table as what you described for the graph in Question 2? 5. Now graph the quadratic function y = x 2. Describe the rate of change in detail below. 6. Study the tables and graphs of y = x 2 and y = x 2. a. What is the major difference between these

8 two functions? b. Does this pattern hold when comparing y = x 2, y = 2x 2, and y = 0.5x2? 7. Let s compare the functions y = x 2, y = 2x 2, and y = 0.5x 2. Graph all three using a standard window. a) Name two things that all three graphs have in common. b) Name two things that all three tables have in common. c) Compare the graphs of y = x 2 and y = 2x 2 --how are they different? d) Compare the graphs of y = x 2 and y = 0.5x 2 -- how are they different? Adding a Constant Another family of quadratic functions are those in the form y = ax 2 + c. 8. Graph y = x 2 and y = x a. Give the coordinates of the y-intercept for y = x 2. b. Give the coordinates of the y-intercept for y = x How is the location of the y-intercept of y = x 2 4 related to the y-intercept of y = x 2? 10. In general, how is the graph of y = ax 2 + c related to the graph of y = ax 2?

9 11. Look at the tables of y = x and y = x 2. a. What are the values of y = x and y = x 2 when x = 0? b. What other relationship exists between the y-values of the two functions tables? 12. In general, how is the table of y = ax 2 + c related to the table of y = ax 2? Adding a Linear Term Another type of a quadratic function is y = ax 2 + b x. 13. What is the difference between the graph of y = x 2 + 4x and the graph of y = x 2? The graph shifts units to the. 14. What is the difference between the graph of y = x 2 4x and the graph of y = x 2? The graph shifts units to the. 15. What is the difference between the graph of y = 2x 2 + 6x and the graph of y = 2x 2? 16. What is the difference between the graph of y = 2x 2 6x and the graph of y = 2x 2? 17. Now, without graphing, what direction do you predict y = x x to shift? a. Graph y = x 2 and y = x x. Were you correct in your prediction? b. Suggest a reason why the graph of y = x x did not follow the pattern you discovered in #13-16.

10 18. How can we determine if a graph will be shifted left or right? (Will do with teacher!) What does the standard form of a quadratic function ax 2 + b x + c tell me about the graph?? 19. The diagram below gives graphs for three of the four quadratic functions listed below. a. Match each function to its graph and state which function is missing. Do not use your graphing calculators. y = x 2 4 x y = x 2 4 x + 6 y = x 2 4 x

11 y = x 2 4x 5 b) Which function s graph is missing? c) Find the axis of symmetry for the function y = x 2 4x using the axis formula x = b. Show work! Lightly draw in the axis on the picture of the graphs above. 2a d) Find the axis of symmetry for the function y = x 2 4 x + 6. e) Find the axis of symmetry for the function y = x 2 4x Label/Draw in the following key parts of a parabola x-intercepts (label both!) y-intercept axis of symmetry vertex

12 Check Your Understanding Use what you know about the relationship between rules and graphs for quadratic functions to match the graphs & functions below. Graphs all have the same windows. Rule Rule Rule Rule Rule Looking at y = ax 2 + b x + c 1) In conclusion the a value tells us two things: 2) In conclusion c value tells us: 3) In conclusion b value tells us: 4) What is the axis of symmetry? How do we determine the axis of symmetry?

13 5) How could we use the axis of symmetry to determine the vertex? Graphing a Quadratic Equation Steps 1) Write the equation in standard form. f (x) = x x 2) Identify the a, b, c values. 3) Find the axis of symmetry. Plot axis of symmetry. 4) Find the vertex. Plot vertex 5) Choose 3 immediate x-values on the right side of the vertex.plug those x-values into the equation. 6) Plot those points 7) Reflect them over the axis of symmetry.

14 8) Connect the dots. Try This Graphing a Quadratic Equation Steps 1) Write the equation in standard form. 2) Identify the a, b, c values. 3) Find the axis of symmetry. Plot axis of symmetry.

15 4) Find the vertex. Plot vertex 5) Choose 3 immediate x-values on the right side of the vertex.plug those x-values into the equation. 6) Plot those points 7) Reflect them over the axis of symmetry. 8) Connect the dots. Check Your Understanding In Game 3 of the 1970 NBA championship series, the L.A. Lakers were down by two points with three seconds left in the game. The ball was inbounded to Jerry West, whose image is silhouetted in today s NBA logo. He launched and made a miraculous shot from beyond midcourt, a distance of 60 feet, to send the game into overtime (there was no 3-point line at that time). Through careful analysis of the game tape, one could determine the height at which Jerry West released the ball, as well as the amount of time that elapsed between the time the ball left his hands and the time the ball reached the basket. This information could be used to write a rule for the ball s height h(t) in feet as a function of time in flight t in seconds. a. Suppose the basketball left West s hands at a point 8 feet above the ground. What are the given variables and their values?

16 b. Suppose also that the basketball reached the basket (at a height of 10 feet) 2.5 seconds after it left West s hands. Use this information to determine the initial upward velocity of the basketball. c. Now that you know the initial upward velocity, write a rule giving h(t) as a function of t. Using Calculator to Find Points of Importance (IAG2)

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