19.1 Understanding Quadratic Functions

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1 Name Class Date 19.1 Understanding Quadratic Functions Essential Question: What is the effect of the constant a on the graph of f () = a? Resource Locker Eplore Understanding the Parent Quadratic Function A function that can be represented in the form of ƒ () = a + b + c is called a quadratic function. The terms a, b, and c, are constants where a. The greatest eponent of the variable is. The most basic quadratic function is ƒ () =, which is the parent quadratic function. A Here is an incomplete table of values for the parent quadratic function. Complete it. B Plot the ordered pairs as points on the graph, and connect the points to sketch a curve. f () = 3 ƒ () = = ( 3) = Houghton Mifflin Harcourt Publishing Compan Reflect 1. Discussion What is the domain of ƒ () =? The curve is called a parabola. The point through which the parabola turns direction is called its verte. The verte occurs at (, ) for this function. A vertical line that passes through the verte and divides the parabola into two smmetrical halves is called the ais of smmetr. For this function, the ais of smmetr is the -ais.. Discussion What is the range of ƒ () =? Module 19 9 Lesson 1

2 Eplain 1 Graphing g () = a when a > The graph g () = a, is a vertical stretch or compression of its parent function ƒ () =. The graph opens upward when a >. Vertical Stretch g () = a with a > 1. The graph of g () is narrower than the parent function ƒ (). Vertical Compression g () = a with < a < 1. The graph of g () is wider than the parent function ƒ (). g() f() f() g() The domain of a quadratic function is all real numbers. When a >, the graph of g () = a opens upward, and the function has a minimum value that occurs at the verte of the parabola. So, the range of g () = a, where a >, is the set of real numbers greater than or equal to the minimum value. Eample 1 Graph each quadratic function b plotting points and sketching the curve. State the domain and range. A g () = g () = Domain: all real numbers Range: - - Houghton Mifflin Harcourt Publishing Compan Module 19 9 Lesson 1

3 B g () = 1 g () = Domain: Range: Reflect 3. For a graph that has a vertical compression or stretch, does the ais of smmetr change? Your Turn Graph each quadratic function. State the domain and range.. g () = 3 5. g () = 1_ Eplain Graphing g () = a when a < Houghton Mifflin Harcourt Publishing Compan The graph of = opens downward. It is a reflection of the graph of = across the -ais. So, When a <, the graph of g () = a opens downward, and the function has a maimum value that occurs at the verte of the parabola. In this case, the range is the set of real numbers less than or equal to the maimum value. Vertical Stretch g () = a with a > 1. The graph of g () is narrower than the parent function f() f() g() Vertical Compression g () = a with < a < 1. The graph of g () is wider than the parent function f (). - g() f() Module Lesson 1

4 Eample Graph each quadratic function b plotting points and sketching the curve. State the domain and range. A g () = g () = Domain: all real numbers Range: B g () = 1_ -3 g () = Domain: Range: 3 Reflect. Does reflecting the parabola across the -ais (a < ) change the ais of smmetr? Houghton Mifflin Harcourt Publishing Compan Module 19 9 Lesson 1

5 Your Turn Graph each function. State the domain and range. 7. g () = 3. g () = Eplain 3 Writing a Quadratic Function Given a Graph You can determine a function rule for a parabola with its verte at the origin b substituting and values for an other point on the parabola into g () = a and solving for a. Eample 3 Write the rule for the quadratic functions shown on the graph. A Use the point (, ). - - (, ) Start with the functional form. g () = a Replace and g () with point values. = a () Evaluate. Divide both sides b to isolate a. = a 1 = a Write the function rule. g () = 1 Houghton Mifflin Harcourt Publishing Compan B Use the point (-, ) (-, -) - - Start with the functional form. g () = a Replace and g () with point values. = a ( ) Evaluate. = a Divide both sides b to isolate a. = a Write the function rule. g () = Module Lesson 1

6 Your Turn (1, ) (1, -1) Eplain Modeling with a Quadratic Function Real-world situations can be modeled b parabolas. Eample A Depth (ards) - For each model, describe what the verte, -intercept, and endpoint(s) represent in the situation it models, and then determine the equation of the function. This graph models the depth in ards below the water s surface of a dolphin before and after it rises to take a breath and descends again. The depth d is relative to time t, in seconds, and t = is when dolphin reaches a depth of ards at the surface (-, -3) (, -3) Time (seconds) The -intercept occurs at the verte of the parabola at (, ), where the dolphin is at the surface to breathe. The endpoint (-, -3) represents a depth of 3 ards below the surface at seconds before the dolphin reaches the surface to breathe. The endpoint (, -3) represents a depth of 3 ards below the surface at seconds after the dolphin reaches the surface to breathe. The graph is smmetric about the -ais with the verte at the origin, so the function will be of the form = a, or d (t) = a t. Use a point to determine the equation. d (t) = a t -3 = a () -3 = a 1 - = a The function is d (t) = - t. Module 19 9 Lesson 1 Houghton Mifflin Harcourt Publishing Compan Image Credits: Malcolm Schul/Alam

7 B Satellite dishes reflect radio waves onto a collector b using a reflector (the dish) shaped like a parabola. (, 1) The graph shows the height h in feet of the reflector 1 relative to the distance in feet from the center of the satellite dish The -intercept occurs at the verte, which represents Distance from Center (feet) the distance = feet from the center of the dish. The left endpoint represents the height h = feet at the center of the dish. The right endpoint represents the height h = feet at the distance = feet from the center of the dish. The function will be of the form. Use (, ) to determine the equation. h () = a = a ( ) Height (feet) 1 = a = 1_ 3 h () = Your Turn 11. The graph shows the height h in feet of a rock dropped down a deep well as a function of time t in seconds. Houghton Mifflin Harcourt Publishing Compan Height (feet) (, -) Time (seconds) Module Lesson 1

8 Elaborate 1. Discussion In eample 1A the points (3, 1) and (-3, 1) did not fit on the grid. Describe some strategies for selecting points used to guide the shape of the curve. 13. Describe how the ais of smmetr of the parabola sitting on the -ais can be used to help plot the graph of ƒ () = a. 1. Essential Question Check-In How can ou use the value of a to predict the shape of ƒ () = a without plotting points? Evaluate: Homework and Practice 1. Plot the function ƒ () = and g () = - on the grid Which of the following features are the same and which are different for the two functions? a. Domain b. Range c. Verte d. Ais of smmetr e. Minimum f. Maimum Online Homework Hints and Help Etra Practice Houghton Mifflin Harcourt Publishing Compan Module 19 9 Lesson 1

9 Graph each quadratic function. State the domain and range.. g () = 3. g () = 1_ g () = 3_ 5. g () = g () = - 1_ 7. g () = - Houghton Mifflin Harcourt Publishing Compan Module Lesson 1

10 . g () = - 3_ 9. g () = Determine the equation of the parabola graphed (-1, 3) (, -) (3, -) (, 5) Houghton Mifflin Harcourt Publishing Compan Module 19 9 Lesson 1

11 A cannonball fired horizontall appears to travel in a straight line, but drops to earth due to gravit, just like an other object in freefall. The height of the cannonball in freefall is parabolic. The graph shows the change in height of the cannonball (in meters) as a function of distance traveled (in kilometers). Refer to this graph for questions 1 and 15. h Height (m) (., -5) d Distance (km) 1. Describe what the verte, -intercept, and endpoint represent. 15. Find the function h (d) that describes these coordinates. Houghton Mifflin Harcourt Publishing Compan Image Credits: (t) Brandon Alms/Alam; (b) Olegusk/Shutterstock A slingshot stores energ in the stretched elastic band when it is pulled back. The amount of stored energ versus the pull length is approimatel parabolic. Questions 1 and 17 refer to this graph of the stored energ in millijoules versus pull length in centimeters. Energ (mj) E (, ) Pull Length (cm) 1. Describe what the verte, -intercept, and endpoint represent. d 17. Determine the function, E (d), that describes this plot. Module Lesson 1

12 Newer clean energ sources like solar and wind suffer from unstead availabilit of energ. This makes it impractical to eliminate more traditional nuclear and fossil fuel plants without finding a wa to store etra energ when it is not available. One solution being investigated is storing energ in mechanical flwheels. Mechanical flwheels are heav disks that store energ b spinning rapidl. The graph shows how much energ is in a flwheel, as a function of revolution speed. 1 E (1, 1) Energ (kwh) 1 Rotation Speed (rps) r 1. Describe what the verte, -intercept, and endpoint represent. 19. Determine the function, E (r), that describes this plot. Phineas is building a homemade skate ramp and wants to model the shape as a parabola. He sketches out a cross section shown in the graph. Height (feet) h (1, ) 5 1 Length (feet). Describe what the verte -intercept, and endpoint represent. l 1. Determine the function, h (l), that describes this plot. Houghton Mifflin Harcourt Publishing Compan Image Credits: Tusumaru/Shutterstock Module 19 9 Lesson 1

13 H.O.T. Focus on Higher Order Thinking. Multipart Classification - - g() - - f() Mark the following statements about ƒ () = and g () = a as true or false. a. a > 1 b. a < c. a > d. a < e. a < 1 f. The graphs of ƒ () and g () share a verte. g. The graphs share an ais of smmetr. h. The graphs share a minimum. i. The graphs share a maimum. 3. Check for Reasonableness The graph of g () = a is a parabola that passes through the point (-, ). Kle sas the value of a must be - 1. Eplain wh this value of a is not reasonable. Houghton Mifflin Harcourt Publishing Compan. Communicate Mathematical Ideas Eplain how ou know, without graphing, what the graph of g () = 1 1 looks like. 5. Critical Thinking A quadratic function has a minimum value when the function s graph opens upward, and it has a maimum value when the function s graph opens downward. In each case, the minimum or maimum value is the -coordinate of the verte of the function s graph. What can ou sa about a when the function ƒ () = a has a minimum value? A maimum value? What is the minimum or maimum value in each case? Module Lesson 1

14 Lesson Performance Task Klie made a paper helicopter and is testing its flight time from two different heights. The graph compares the height of the helicopter during the two drops. The graph of the first drop is labeled g () and the graph of the second drop is labeled h (). a. At what heights did Klie drop the helicopter? What is the helicopter s flight time during each drop? b. If each graph is represented b a function of the form ƒ () = a, are the coefficients positive or negative? Eplain. c. Estimate the functions for each graph. Height (ft) g() Helicopter s Height h() Time (s) Houghton Mifflin Harcourt Publishing Compan Module 19 9 Lesson 1