ONLINE PAGE PROOFS. Quadratic functions Overview TOPIC 17. Why learn this? What do you know? Learning sequence. number and algebra

Transcription

1 TOPIC 7 Quadratic functions 7. Overview Wh learn this? A quadratic function can be used to describe a phsical thing such as the arch of a bridge, the path of a batted ball, water rising from a fountain, or a whale s spout. Over the centuries the quadratic equation has plaed a major role in the whole of human civilisation as we know it. We would not be able to watch satellite television or engage in other modern activities without the developments that have come about through the application of quadratic functions. What do ou know? THInK List what ou know about quadratic graphs. Use a thinking tool such as a concept map to show our list. PaIr Share what ou know with a partner and then with a small group. SHare As a class, create a thinking tool such as a large concept map to show our class s knowledge of quadratic graphs. Learning sequence 7. Overview 7. Graphs of quadratic functions 7. Plotting points to graph quadratic functions 7. Sketching parabolas of the form = a 7. Sketching parabolas of the form = a + c 7.6 Sketching parabolas of the form = ( h) 7.7 Sketching parabolas of the form = ( h) + k 7.8 Sketching parabolas of the form = ( + a)( + b) 7.9 Applications 7. Review ONLINE ONLY

2 WaTCH THIS VIdeO The stor of mathematics: Catapults and projectiles Searchlight Id: eles-7

3 7. Graphs of quadratic functions The graph at right is a tpical parabola with features as listed below. The dotted line is the ais of smmetr; the parabola is the same on either side of this line. The turning point is the lowest point on the graph; the point where the graph changes direction. It is also called the local minimum. The parabola is upside down, or inverted, if the turning point is the highest point on the graph. The -intercept(s) is where the graph crosses (or sometimes just touches) the -ais. Not all parabolas have -intercepts. The -intercept is where the graph crosses the -ais. All parabolas have one -intercept. WOrKed example For each of the following graphs, state the equation of the ais of smmetr, the coordinates of the turning point and whether it is a maimum or a minimum. a Ais of smmetr b Ais of smmetr THInK (, ) a State the equation of the vertical line that cuts the parabola in half. (, ) WrITe a Ais of smmetr is =. State the turning point. The turning point is at (, ). Determine the nature of the turning point b observing whether it is the highest or lowest point of the graph. b State the equation of the vertical line that cuts the parabola in half. Minimum turning point b Ais of smmetr is =. State the turning point. The turning point is at (, ) intercept (, ) -intercepts (, ) (, ) = ( ) Local minimum (, ) Ais of smmetr = Determine the nature of the turning point b observing whether it is the highest or lowest point of the graph. Maimum turning point 6 Maths Quest 9

4 The - and -intercepts The -intercept is where the graph crosses (or just touches) the -ais. The -intercept is where the graph crosses the -ais. All parabolas have one -intercept. When sketching a parabola, the -intercepts (if an) and the -intercept should alwas be marked on the graph, with their respective coordinates. WOrKed example For each of the following graphs, state the equation of the ais of smmetr, the coordinates of the turning point, whether the point is a maimum or a minimum, and the - and -intercepts. a THInK (, ) a State the equation of the vertical line that cuts the parabola in half. State the turning point and its nature; that is, determine whether it is the highest or lowest point of the graph. Observe where the parabola crosses the -ais. In this case, the graph touches the -ais when =, so there is onl one -intercept. Observe where the parabola crosses the -ais. b State the equation of the vertical line that cuts the parabola in half. State the turning point and its nature; that is, determine whether it is the highest or lowest point of the graph. Observe where the parabola crosses the -ais. Observe where the parabola crosses the -ais. b (, ) WrITe a Ais of smmetr is =. Maimum turning point is at (, ). The -intercept is. It occurs at the point (, ). The -intercept is. It occurs at the point (, ). b Ais of smmetr is =. Minimum turning point is at (, ). The -intercepts are and. The occur at the points (, ) and (, ). The -intercept is. It occurs at the point (, ). Topic 7 Quadratic functions 6

5 Eercise 7. Graphs of quadratic functions IndIVIdual PaTHWaYS reflection What are the major features of all parabolas? PraCTISe Questions: 8,, 6, 7 COnSOlIdaTe Questions: a, c, e, a, c, e,,, 6, 7, 9,, 7 master Questions: a, c, f, a, f,,, 7, 9 9 Individual pathwa interactivit int- doc 989 FluenCY WE For each of the graphs below: i state the equation of the ais of smmetr ii give the coordinates of the turning point iii indicate whether it is a minimum or maimum turning point. a b c (, ) (, ) (, ) d e f (, ) (, ) (, ) For each of the graphs below, state: i the equation of the ais of smmetr ii the coordinates of the turning point iii whether the turning point is a maimum or minimum. a b c (, ) (, ) (, ) (, ) d e f (, ) (, ) (, 6) (, ) (, ) 6 Maths Quest 9

6 WE For each of the following graphs, state the equation of the ais of smmetr, the coordinates of the turning point and whether it is a maimum or a minimum, and the - and -intercepts. a b (, ) c d g j 9 (, ) e h MC a The ais of smmetr for the graph shown at right is: A = B = C = D the -ais b The coordinates of the turning point for the graph are: A (, ) B (, ) C (, ) D (, ) c The -intercept is: A B C D d The -intercepts are: A and B and C and D and f i Topic 7 Quadratic functions 6

7 UNDERSTANDING Consider the table of values below. a Plot these points on graph paper. What shape is the graph? b Locate the ais of smmetr. c Locate the -intercept. d Locate the -intercept(s). 6 Consider the function = +. Complete this table of values for the function MC Which of the following rules is not a parabola? A = B = C = D = + 8 Consider the graph of =. a State the turning point of this graph. b State whether the turning point is a maimum or a minimum. 9 Given the following information, make a sketch of the graph involved. a Maimum turning point = (, ), -intercept = (, 6) b Minimum turning point = (, ), -intercepts (, ) and ( 7, ) REASONING Consider the parabola given b the rule = and the straight line given b =. Show that the two graphs meet at (, ) and (, ). A window-cleaning compan varies its charges as the square of the height of the building. Let the heights of buildings be,,,... m. The cost of window washing for a -m building is $. a Determine the window washing costs for the buildings listed above. B plotting a graph, what shape is the graph? b What would be the cost for a -m building? c If the cost is $79, show that the height of the building is 7 m. Another window-cleaning compan also varies its charges as the square of the height of the building. The cost for a -m building is $6. a Show that the cost for a -m building is $. b If the cost was $., what was the height of the building? If the ais of smmetr of a parabola is = and one of the -intercepts is (, ), show the other -intercept is ( 8, ). If the -intercepts of a parabola are (, ) and (, ), show that the ais of smmetr is =.. PROBLEM SOLVING On a set of aes, sketch a parabola that has no -intercepts and has an ais of smmetr =. Can the parabola have a maimum or minimum turning point or both? Eplain our reasoning. 6 Sketch a graph where the turning point and the -intercept are the same. Suggest a possible equation. 6 Maths Quest 9

8 7 Sketch a graph where the turning point and the -intercept are the same. Suggest a possible equation. 8 a Complete the table to show how man intersections there are for,,, and 6 lines. n (number of lines) 6 N (number of intersections) b Find the rule connecting the number of lines, n, with the number of intersections, N, and hence show that the relationship is quadratic. c If ou could draw lines, how man intersections would there be? 7. Plotting points to graph quadratic functions If there is a rule connecting and, a table of values can be used to determine actual coordinates. When drawing straight line graphs, a minimum of two points is required. For parabolas there is no minimum number of points, but between 6 and points is a reasonable number. The more points used, the smoother the parabola will appear. The points should be joined with a smooth curve, not ruled. Ensure that points plotted include (or are near) the main features of the parabola, namel the ais of smmetr, the turning point and the - and -intercepts. WOrKed example Cop and complete the table of values for each of the following equations, then list the coordinates of each of the points. a = = 9 b = + = + c = + = + THInK WrITe/dISPlaY a Write the equation. a = Substitute the -values into the equation to obtain the corresponding -values. When =, = ( ) = 9 When =, = ( ) =... When =, = () = 9 Topic 7 Quadratic functions 6

9 Complete the table of values. = 9 9 List the coordinates of each of the points. (, 9), (, ), (, ), (, ), (, ), (, ), (, 9) b Write the equation. b = + Substitute the -values into the equation to obtain the corresponding -values. Complete the table of values. When =, = ( ) + = 7 When =, = ( ) + = When =, = () + = 7 = List the coordinates of each of the points. (, 7), (, ), (, ), (, ), (, ), (, ), (, 7) c Write the equation. c = + Substitute the -values into the equation to obtain the corresponding -values. Complete the table of values. When =, = ( ) ( ) + = 9 When =, = ( ) ( ) + = 8... When =, = () () + = = List the coordinates of each of the points. (, 9), (, 8), (, ), (, ), (, ), (, ), (, ) WOrKed example Occasionall a list of -values will be provided and the corresponding -values can be calculated. In the following eample, the set of -values is specified as. Plot the graph of = +, and, hence, state: a the equation of the ais of smmetr b the coordinates of the turning point and whether it is a maimum or a minimum c the - and -intercepts. THInK WrITe/dISPlaY Write the equation. = + Complete a table of values b substituting into the equation each integer value of from to. For eample, when =, = ( ) + ( ) =. List the coordinates of the points. (, ), (, ), (, ), (, ), (, ), (, ), (, ) 66 Maths Quest 9

10 Draw and label a set of aes, plot the points listed and join the points to form a smooth curve. = + (, ) a b Find the equation of the line that divides the parabola eactl into two halves. Find the point where the graph turns or changes direction, and decide whether it is the highest or lowest point of the graph. State the coordinates of this point. c State the -coordinates of the points where the graph crosses the -ais. State the -coordinate of the point where the graph crosses the -ais. a Ais of smmetr is =. b Minimum turning point is at (, ). c The -intercepts are at and. The occur at the points (, ) and (, ). The -intercept is at. It occurs at the point (, ). A rule connecting and will be occasionall provided. From this rule, pairs of - and -values can be calculated. In the following eample, the rule is given as h = +. Graphs can be drawn using a graphics calculator, a graphing program on our computer or b hand. WOrKed example Rudie, the cannonball chicken, was fired out of a cannon. His path could be traced b the equation h = +, where h is Rudie s height, in metres, above the ground and is the horizontal distance, in metres, from the cannon. Plot the graph for and use it to find the maimum height of Rudie s path. THInK WrITe Write the equation. h = + Complete a table of values b substituting into the equation each integer value of from to. For eample, when =, h = + =. h List the coordinates of the points. (, ), (, ), (, ), (, ), (, ), (, ) Topic 7 Quadratic functions 67

11 As a parabola is smmetrical, the greatest value of h must be greater than and occurs when lies between the and, so find the value of h when =.. Draw and label a set of aes, plot the points from the table and join the points to form a smooth curve. When =., h =. +. =. h h = + 6 The maimum height is the value of h at the highest point of the graph. h =. 7 Answer the question in a sentence. The maimum height of Rudie s path is. metres. reflection If given a choice, what is the best wa to choose the points to be plotted on a graph? doc-99 doc-99 Eercise 7. Plotting points to graph quadratic functions IndIVIdual PaTHWaYS PraCTISe Questions:,,, a, c, e, g, 7,, COnSOlIdaTe Questions:,,, b, d, f, h,, 6, 8,,,, master Questions:,,, a, f, g,, 6, 9 7 FluenCY WE Cop and complete the table of values for each of the following equations, then list the coordinates of each of the points. a = = b = = c = = + + Individual pathwa interactivit int- Using the equations in question : i plot the points and join with a smooth curve ii identif the ais of smmetr and state its equation. 68 Maths Quest 9

12 Complete the following table of values, plot the points, and then join with a smooth curve. = + WE Plot the graph of each of the following and, hence, state: i the equation of the ais of smmetr ii the coordinates of the turning point and whether it is a maimum or a minimum iii the - and -intercepts. Remember: 7 means the graph is drawn from = 7 to =. a = + 8 +, 7 b =, c =, d = +, e = + +, 9 f = + +, 6 g = 6 8, 9 h =, Consider the equations for : i = + ii = +. a Make a table of values and plot the points on the same set of aes. b State the equation of the ais of smmetr for each equation. c State the -intercepts for each equation. UNDERSTANDING 6 WE A missile was fired from a boat during a test. The missile s path could be traced b the equation h = +, where h is the missile s height above the ground, in kilometres, and is the horizontal distance from the boat, in kilometres. Plot the graph for and use it to find the maimum height of the missile s path, in metres. 7 The speed versus distance graph of a car braking efficientl has the equation s = v, where v is the speed, in km/h, and s is the stopping distance, in metres. 6 a Use graph paper to plot this graph for v (choose an appropriate scale). From the graph find the stopping distance for a car at: i 6 km/h ii km/h iii km/h. b What is the maimum speed a car can travel if it must stop within m? Round answers to the nearest whole number. Compare our answers from the graph and formula. 8 SpaceCorp sent a lander to Mars to measure the temperature change over a period of time. The results were plotted on a set of aes shown below. From the graph it can be seen that the temperature change follows the quadratic rule T = h + h, where T is the temperature in degrees Celsius, and the time elapsed, h, is in hours. Topic 7 Quadratic functions 69

13 T 8 6 a What was the initial temperature on Mars? b When was the temperature measured as C? c When was the highest temperature recorded? d What was the highest temperature recorded? 9 A ball was thrown from the top of a building. Its height above the ground, h, is given b h =.9t + t +., where h is in metres and t is the time, in seconds, since the ball was thrown. a What is the height of the building? (Hint: Where was the ball at t =?) b Sketch a graph of h versus t. c How long did it take for the ball to reach the ground? REASONING The Grand Old Duke of York marched his men up a hill that follows the path of the equation = + 6 where is the vertical distance travelled and is the horizontal distance. Both measurements are in metres. Plot the graph for 6, and use it to find out whether the Duke and his men were halfwa across when the were halfwa up. If not, eplain how far across the were on the upward journe. Round answers to decimal place. When a golfer hits the ball from the tee with a 7 iron, the ball follows the path of the parabola h = When hit with a 9 iron, the ball follows the path of the parabola h = In both cases, h is the height above ground, in metres, and is the horizontal distance from the golfer, in metres. The green is 8 metres awa from the tee. Plot both graphs to determine which club the golfer should use. On a basketball court, Sam threw a basketball towards the ground such that it followed the path defined b the equation =.( ), where is the height of the ball in metres and is the horizontal distance from Sam in metres. a How far off the ground was the ball when Sam threw it? b How far from Sam did the ball bounce? c If Sane, another plaer, is metres awa from Sam, eplain how high he will need to jump in order to catch the ball. (Sane is.8 m tall and his arm length is 7 cm.) In order for Rub to participate in a trip to the snow this ear, she will need to fund-raise. B selling mini Easter eggs at school for different prices each da, Rub found that the h 6 Maths Quest 9

14 relationship between the cost (c) of a sachet of mini Easter eggs and the total profit (P) was modelled b the equation P = (c + )(6 c). One da she did not sell an Easter eggs but a teacher kindl gave her $ towards her trip. a Draw a graph of P = (c + )(6 c). b Eplain wh the graph should not appear to the left of the P ais. c What will the mini Easter eggs cost if the maimum profit is to be made? How much will this profit be? d Up to what amount could Rub charge before she makes a loss? PrOblem SOlVIng a The ais of smmetr of a parabola is =. If one -intercept is, what is the other -intercept? b Suggest a possible equation for the parabola. a Compare the turning points of the parabolas = 6 and = 6. b Find an equation for a quadratic curve that has the same ais of smmetr as = 6 and a turning point (.,.). 6 Water comes out of a garden hose in the shape of a parabola and can be modelled b the equation = , where is the horizontal distance from the hose s nozzle and is the height of the water (both in metres). Investigate whether the water can be jetted over a -metre fence that is located. metres from the nozzle. 7 The data below models the equation of = a + b + c. Use the table to show that c = and find the values of a and b Sketching parabolas of the form = a The graph of the quadratic function = The simplest parabola, =, is shown at right. Both the - and -aes are clearl indicated, along with their scales. The turning point (, ) is indicated. The - and -intercepts are indicated. For this graph the are all (, ). This is an eample of a parabola that just touches (does not cross) the -ais at (, ). = Parabolas of the form = a, where a > A coefficient in front of the term affects the dilation of the graph, making it wider or narrower than the graph of =. If a > then the graph becomes narrower, whereas if < a <, the graph becomes wider. The following features of the parabola remain unchanged, for a >, regardless of the value of a: the ais of smmetr is = the turning point is (, ) the -intercept is (, ) the -intercept is (, ) the shape of the parabola is alwas upright or a U shape. (, ) int-79 Topic 7 Quadratic functions 6

15 WOrKed example 6 On the same set of aes sketch the graph of = and =, marking the coordinates of the turning point and the intercepts. State which graph is narrower. THInK Write the equation of the first graph. = WrITe/draW State its ais of smmetr. The ais of smmetr is =. State the coordinates of the turning point. The turning point is (, ). State the intercepts. The -intercept is and the -intercept is. Find the coordinates of one other point. When =, =. (, ) 6 Write the equation of the second graph. = 7 State its ais of smmetr. The ais of smmetr is =. 8 State the coordinates of the turning point. The turning point is (, ). 9 State the intercepts. The -intercept is and the -intercept is. Find the coordinates of one other point. When =, =. (, ) Sketch the graphs, labelling the turning point. = (, ) = State which graph is narrower. The graph of = is narrower. Parabolas of the form = a, where a < When a <, the graph is inverted; that is, it is shaped. A coefficient in front of the term affects the dilation of the graph, making it wider or narrower than the graph of =. If < a <, the graph is wider than =. If a <, the graph is narrower than =. The following features of the parabola remain unchanged, for a <, regardless of the value of a: the ais of smmetr is = the turning point is (, ) the -intercept is (, ) the -intercept is (, ) The shape of the parabola is alwas inverted or an upside-down U shape ( ). 6 Maths Quest 9

16 WOrKed example 7 On the same set of aes sketch the graphs of = and =, marking the coordinates of the turning point and the intercept. State which graph is narrower. THInK WrITe/draW Write the equation of the st graph. = State its ais of smmetr. The ais of smmetr is =. State the coordinates of the turning point. The turning point is (, ). State the intercepts. The -intercept is (, ) and the -intercept is also (, ). Calculate the coordinates of one other point. When =, =, (, ). 6 Write the equation of the nd graph. = 7 State its ais of smmetr. The ais of smmetr is =. 8 State the coordinates of the turning point. The turning point is (, ). 9 State the intercepts. The -intercept is (, ) and the -intercept is also (, ). Calculate the coordinates of one other point. When =, =, (, ). Sketch the two graphs on a single set of aes, labelling the turning point (as well as intercepts and maimum). (, ) = = The graph of = is narrower. Eercise 7. Sketching parabolas of the form = a IndIVIdual PaTHWaYS PraCTISe Questions: 7, 8,,, 6 COnSOlIdaTe Questions: 7, 9,,,, 6 8 Individual pathwa interactivit int-6 master Questions: 7,, reflection List the features of a parabola that remain unchanged when a changes from positive to negative in = a. FluenCY WE6 On the same set of aes, sketch the graph of = and =, marking the coordinates of the turning point and the intercepts. State which graph is narrower. On the same set of aes sketch the graph of = and =, marking the coordinates of the turning point and the intercepts. State which graph is narrower. Topic 7 Quadratic functions 6

17 Sketch the graph of the following table. State the equation of the graph..... WE7 Using the same set of aes, sketch the graphs of = and =., marking the coordinates of the turning point and the intercepts. State which graph is narrower. Using the same set of aes from question, sketch the graph of =, marking the coordinates of the turning point and the intercepts. State which graph is the narrowest. 6 MC a The graph of = is: A wider than = B narrower than = C the same width as = b The graph of = is: D a reflection of = in the -ais A wider than = B narrower than = C the same width as = c The graph of = is: A wider than = C the same width as = D a reflection of = in the -ais B narrower than = D a reflection of = in the -ais 7 Match each of the following parabolas with the appropriate equation from the list. i = ii = iii = iv = v = vi = a d b e c f UNDERSTANDING 8 Write an equation for a parabola that has a minimum turning point and is narrower than =. 9 Write an equation for a parabola that has a maimum turning point and is wider than =. Find the equation of a quadratic relation if it has an equation of the form = a and passes through: a (, ) b (, ). Consider the equation =.. Calculate the values of when is: a b c d. e.. 6 Maths Quest 9

18 The amount of power (watts) in an electric circuit varies as the square of the current (amperes). If the power is watts when the current is amperes, calculate: a the power when the current is amperes b the power when the current is amperes. REASONING a Sketch the following graphs on the same aes: =, = and =. Shade the area between the two graphs above the -ais and the area inside the graph below the -ais. Describe the shape that has been shaded. b Sketch the following graphs on the same aes: =, = and =. Shade the area inside the graphs of = and =. Also shade the area between the graph of = and the -ais. Describe the shape that ou have drawn. This figure at right shows the parabolic shape of a skate ramp. The rule of the form = a describes the shape of the ramp. If the top of the ramp has coordinates (, 6) find a possible equation that describes the shape. Justif our answer. The total sales of a fast food franchise varies as the square of the number of franchises in a given cit. Let S be the total sales (in millions of dollars per month) and f be the number of franchises. If sales = $ million when f =, then: a show that the equation relating S and f is S = 6 f. b determine the number of franchises needed to (at least) double the sales from $. PROBLEM SOLVING 6 Xanthe and Carl were comparing parabolas on their CAS calculators. Xanthe graphed =. with window settings and. Carl graphed = with window settings k k and k. Ecept for the scale markings, the graphs looked eactl the same. What is the value of k? 7 The parabola = is rotated 9 clockwise about the origin. Find the equation. 8 Determine which pairs of the following parabolas are congruent. =, =, =, = +, =, = 9 Letting a >, use values of a to show that ever parabola = a + b + c is congruent to = a. 7. Sketching parabolas of the form = a + c Adding a constant, c, to the rule = a translates the graph verticall. If c >, the value of increases, therefore the graph translates verticall upwards. If c <, the value of decreases, therefore the graph translates verticall downwards. Topic 7 Quadratic functions 6

19 WOrKed example 8 For each part of the question, sketch the graph of =, then, on the same aes, sketch the given graph, clearl labelling the turning point. a = + b = THInK a Sketch the graph of = b drawing a set of labelled aes, marking the turning point (, ) and noting that it is smmetrical about the -ais. Find the turning point of = + b adding to the -coordinate of the turning point of =. Using the same aes as for the graph of =, sketch the graph of = +, marking the turning point and making sure that it is the same width as the graph of =. (The coefficient of is the same for both graphs.) b Sketch the graph of = b drawing a set of labelled aes, marking the turning point (, ) and noting that it is smmetrical about the -ais. Find the turning point of = b subtracting from the -coordinate of the turning point of =. Using the same aes as for the graph of =, sketch the graph of =, marking the turning point, inverting the graph and making sure that the graph is the same width as the graph of =. WrITe/draW a b = (, ) The turning point of = + is (, ). (, ) (, ) = (, ) = + = The turning point of = is (, ). (, ) = (, ) = If the graph is of the form = + c and c >, the graph translates verticall upwards, and if c <, the graph translates verticall downwards. 66 Maths Quest 9

20 WOrKed example 9 Sketch the graph of = +, drawing clearl labelled aes and marking the ais of smmetr, turning point and -intercept. State whether the turning point is a maimum or minimum. THInK The equation of the ais of smmetr is the same as for =. The turning point has been moved up c units and is (, c). The coefficient of is negative so the graph is inverted. Draw clearl labelled aes, mark the turning point and draw the graph. WrITe/draW The ais of smmetr is =. The turning point is (, ). Maimum turning point (, ) Eercise 7. Sketching parabolas of the form = a + c IndIVIdual PaTHWaYS PraCTISe Questions: 7, 8,,, COnSOlIdaTe Questions: 7, 8,,, 7 Individual pathwa interactivit int-7 master Questions: 7, 8,, 9 = + FluenCY WE8 For each part of the question, sketch the graph of =, then, on the same aes, sketch the given graph, clearl labelling the turning point. a = + b = + c = d = e = + f = How does a positive number at the end of the equation affect the graph? How does a negative number at the end of the equation affect the graph? WE9 Sketch each of the following graphs on clearl labelled aes, marking the ais of smmetr, turning point and -intercept of each one. State whether the turning point is a maimum or a minimum. a = + b = c = + d = + reflection Describe the effect of changing a and c, separatel, in the rule = a + c. int-9 e = f = Topic 7 Quadratic functions 67

21 Sketch the following graphs, indicating the turning point and estimating the -intercepts. a = b = c = 6 a Does the turning point change if there is a negative number in front of the term in the equation =? b How does a negative coefficient of affect the graph? c What is the ais of smmetr for all the graphs in this eercise? 7 MC a The turning point for the graph of the equation = + 8 is: A (, ) B (, 8) C (, 8) D (, 8) b The turning point of the graph of the equation = 6 is: A (, 6) B ( 6, ) C (, ) D (, 6) c The graph of = 7 moves the graph of = in the following wa: A up B down C up 7 D down 7 d The -intercept of the graph of = 6 is: A B C 6 D 6 UNDERSTANDING 8 Match each of the following parabolas with the appropriate equation from the list. i = ii = + iii = iv = + v = + vi = a d b e c f The vertical cross-section through the top of the mountain called the Devil s Tower can be approimated b the graph = +. Sketch the graph. If the -ais represents sea level, and both and are in kilometres, find the maimum height of the mountain. The cross-section of a large bowl can be given b the rule =, where both (measured across the bowl) and (the depth of the bowl) are measured in centimetres. a B factorising the rule, find the points where =. What are these points called? b If the bowl s rim occurs at the point where =, find the greatest depth of the bowl. c What is the width of the bowl at its rim? 68 Maths Quest 9

22 number and algebra The photo at right shows an imaginar line drawn across the surface of a lake. A vertical cross-section of the lake is taken at the line, such that the depth of the lake can be approimated b the graph =, where and are in metres. a What would be a suitable domain of -values for this graph? b Sketch the graph over this domain. c What is the greatest depth of the lake? d What is the width of the lake along the white line shown in the photo? reasoning Show that the equation of the parabola that is of the form = + c and passes through: a (, ) is = b (, ) is =. A ball is thrown verticall upwards and follows a path given b h = at + c, where h is the height of the ball in metres and t is the time of flight in seconds. If the ball reached a height of m and took seconds to reach this height and another seconds to reach the ground again, show that the rule for the flight path of the ball is h = t + 6. Sketch = and = + on the same set of aes. Use algebra to eplain where the intersect. PrOblem SOlVIng The path of a ball rolling off the end of a table follows a parabolic curve and can be modelled b the equation = a + c. A student rolls a ball off a tabletop that is 8 cm above the floor, and the ball lands 8 cm awa horizontall. If the student sets a cup 78 cm above the floor to catch the ball in mid-flight, where should the cup be placed? 6 On a set of aes, plot the points (, ), (, ), (, ) and (, ). On the same set of aes, reflect the points about the -ais and plot the points. Join all the points together and find the equation. 7 What is the meaning of the intersection point between the graphs with equations = and = 6 +? 8 For the equation =, find the eact value of the -coordinate when the -coordinate is + ". doc-9 Topic 7 Quadratic functions c7quadraticfunctions.indd /7/ :9 AM

23 7.6 Sketching parabolas of the form = ( h) Subtracting a constant value h from = translates the parabola h units to the right; otherwise the parabola s shape is unchanged. The equation is = ( h), where h is a positive quantit. The -intercept occurs when = h. Adding a constant value h to in = translates the parabola h units to the left; otherwise the parabola s shape is unchanged. The equation is = ( + h), where h is a positive quantit. The -intercept occurs when = h. The -intercept occurs when =, and is alwas at = h. WOrKed example On clearl labelled aes, sketch the graph of = ( ), marking the turning point and -intercept. State whether the turning point is a maimum or a minimum. THInK WrITe/draW Write the equation. = ( ) Find the ais of smmetr ( = h where h is ). Find the turning point, which has been moved to the right. The sign in front of the bracket is positive so the parabola is upright. Substitute for in the equation to find the -intercept. 6 Draw a clearl labelled set of aes, mark the turning point and -intercept and draw the graph of = ( ). Note that the sign in the brackets is negative so the graph moves units to the right. The ais of smmetr is =. The turning point is (, ). Minimum turning point -intercept: when =, = ( ) = The -intercept is. = ( ) Observe that the turning point also shifts b the same amount and direction as the -intercept. 66 Maths Quest 9

24 WOrKed example Sketch the graph of = ( + ), labelling the turning point, stating whether it is a maimum or minimum, finding the -intercept and the equation of the ais of smmetr. THInK WrITe/draW Write the equation. = ( + ) Find the turning point, which has been moved to the left. The sign in front of the bracket is negative so the parabola is inverted. The -intercept is where =, so substitute for in the equation. Find the ais of smmetr ( = h, where h is ). 6 Draw a clearl labelled set of aes, mark the turning point and -intercept and draw the graph of = ( + ). The turning point is (, ). Maimum turning point -intercept: when =, = ( + ) = The -intercept is. The ais of smmetr is =. = ( + ) Eercise 7.6 Sketching parabolas of the form = ( h) IndIVIdual PaTHWaYS PraCTISe Questions: 7, 8,,, COnSOlIdaTe Questions: 7, 8,,,, 6 Individual pathwa interactivit int-8 master Questions: 7, 8, 9a, a, c, 8 reflection What changes and what remains the same when a changes from a positive to a negative number in = a( h)? FluenCY WE On clearl labelled aes, sketch the graphs of each of the following, marking the turning point and -intercept. State whether the turning point is a maimum or a minimum. a = ( ) b = ( ) c = ( + ) d = ( + ) e = ( ) f = ( + 6) State the ais of smmetr for each of the graphs in question. int-9 Topic 7 Quadratic functions 66

25 How does a positive number for h in = ( h) affect the graph of =? How does a negative number for h in = ( h) affect the graph of =? WE On clearl labelled aes, sketch the graphs of each of the following, marking the turning point and -intercept. State whether the turning point is a maimum or a minimum. a = ( ) b = ( ) c = ( + ) d = ( + ) e = ( ) f = ( + 6) 6 Do the answers to questions and change if there is a negative sign in front of the bracket? How does this negative sign affect the graph? 7 MC a The ais of smmetr for the graph = ( + ) is: A = B = C = D = b The turning point of the graph = ( + ) is: A (, ) B (, ) C (, ) D (, ) UNDERSTANDING 8 Match each of the following parabolas with the appropriate equation from the list. i = ( ) ii = iii = ( + ) iv = ( + ) v = ( ) vi = ( ) a b c d e f 9 Find the equation of the parabola that is of the form = ( h) and passes through: a (, ) b (, 9). State the differences, if an, between the following pairs of parabolas. a = ( ) and = ( ) b = ( ) and = ( + ) c = ( ) and = ( ) Write the rule for the graph in the form = a( h), such that the -intercept is = and the -intercept is at =. 66 Maths Quest 9

26 REASONING The figure below shows the span of the Gateshead Millennium Bridge in England. A set of aes has been superimposed onto the photo. Use the coordinates (, ) and (, 7) to show that a possible equation is =. 87 ( ). 7 a B sketching the graph, or using another method, determine the - and -intercepts of = ( ). b Using the result from part a, show that the -intercept and -intercept for the equation = a( h) are (h, ) and (, ah ) respectivel. The amount of profit made b a new compan each month can be modelled b the rule = a(t h), where is the monthl profit and t is the integer value of the month ( = Januar). If the Januar profit was $9 and the March profit was : a determine the values of a and h b determine the month when the profit is $ c eplain wh normall three points are required to uniquel determine a parabola, but in this eample onl two points are required. PROBLEM SOLVING The graph of a quadratic function touches the -ais and has its turning point at (, ). Graph two parabolas that appl to this description and find equations for them. How man eamples are possible? Eplain. 6 Sketch the graphs of = ( ) and = ( ). What do ou notice about the graphs? Eplain wh this is true. 7 Find the equation of a parabola whose turning point is (, ) and is the same size as the graph = ( ). Topic 7 Quadratic functions 66

27 7.7 Sketching parabolas of the form = ( h) + k The equation = ( h) + k combines a vertical translation of k and a horizontal translation of h together. The equation = ( h) + k is called turning point form because the turning point is given b the coordinates (h, k). WOrKed example Sketch the graph of = ( + ), marking the turning point and the -intercept, and indicate the tpe of turning point. THInK WrITe/draW Write the equation. = ( + ) State the turning point. As the equation is The turning point is (, ). in the form = ( h) + k, the turning point is (h, k). There is no sign outside the brackets, so Minimum turning point the parabola is upright. Find the -intercept b substituting = into the equation. -intercept: when =, = ( + ) = Draw clearl labelled aes, mark the coordinates of the turning point, the -intercept and draw a smmetrical graph. = ( + ) (, ) Note that in the previous eample the graph has shifted units left (h = ) and unit down (k = ). The graph has the same shape as =. WOrKed example On the same set of aes, sketch the graphs of each of the following, clearl marking the coordinates of the turning point and the -intercept: i = ii = ( ) iii = ( ) + State the changes that are made from i to ii and from i to iii. THInK i Sketch the graph of =, marking the coordinates of the turning point and the -intercept. WrITe/draW i = (, ) 66 Maths Quest 9

28 ii Write the equation. ii = ( ) Find the coordinates of the turning point. The turning point is (, ). Find the -intercept. -intercept: when =, = ( ) = On the same set of aes, sketch the graph of = ( ), marking the coordinates of the turning point and the -intercept. = = ( ) (, ) iii Write the equation. iii = ( ) + Find the coordinates of the turning point. (, ) The turning point is (, ). Find the -intercept. -intercept: when = = ( ) + = + = On the same set of aes, sketch the graph of = ( ) +, marking the coordinates of the turning point and the -intercept. State how = is changed to form = ( ). State how = is changed to form = ( ) +. = = ( ) + (, ) = ( ) If = is moved units to the right, the resulting graph is = ( ). If = is moved units to the right and unit up, the resulting graph is = ( ) +. Eercise 7.7 Sketching parabolas of the form = ( h) + k IndIVIdual PaTHWaYS PraCTISe Questions: 8,,,, 6 COnSOlIdaTe Questions:,, a, c, e, 8,,, 6 8 master Questions:,, a, e, f,, reflection What are the advantages of having the equation of a parabola in turning point form? Individual pathwa interactivit int-9 Topic 7 Quadratic functions 66

29 int-8 FluenCY WE Sketch the graph of each of the following, marking the turning point, the tpe of turning point and the -intercept. a = ( ) + b = ( + ) c = ( ) + d = ( + ) e = ( + ) f = ( ) + Eplain wh the equation presented in the form = ( h) + k is known as turning point form. WE Using the same set of aes, sketch each (a f) of the following sets (i, ii, iii) of graphs, clearl marking the coordinates of the turning point and the -intercept. State the changes that are made from i to ii and from i to iii. a i = ii = ( ) iii = ( ) + b i = ii = ( ) iii = ( ) c i = ii = ( + ) iii = ( + ) + d i = ii = ( ) iii = ( ) e i = ii = ( + ) iii = ( + ) f i = ii = ( ) iii = ( ) MC a For the graph of = ( + ), the coordinates of the turning point are: a (, ) b (, ) C (, ) d (, ) b For the graph of = ( ) + 7, the ais of smmetr is: a = b = 7 C = d = c For the graph of = ( ), the -intercept is: a b C d The graph of = is translated units to the left and units up. a State the equation of the translated graph. b State the location of the turning point. 6 The turning point (maimum) of a graph is (, ). a State the equation of the graph in turning point form. b Calculate the -intercept. 7 Sketch the graph that is the mirror image (reflected verticall) of = ( ) +. understanding 8 State the equation of each of the following given it is of the form = ( h) + k or = ( h) + k. a b c Maths Quest 9

30 d 6 6 e f 6 9 Aravind threw a ball into the air. It followed the path of a parabola defined b the equation h = ( ) +, where h metres represents the height of the ball above the ground metres horizontall from Aravind s hand. a Sketch the graph showing the path of the ball during its flight from Aravind s hand until it reaches the ground, marking on the graph the turning point and the -intercept. b Use the graph to find the height of the ball when it leaves Aravind s hand. c Use the graph to find the maimum height that the ball reaches. Write the equations for the following transformations on =. a Reflection in the -ais and translation of units to the left b Translation of unit down and units to the right c Translation of unit left and units up d Reflection in the -ais, translation of units to the right and 6 units up A rocket is shot in the air from a given point (, ). It reaches a maimum height of m and lands m awa from the launching point. a State the equation of the path of the rocket. b How far horizontall from the launching point does the rocket reach its maimum height? MC A parabola has an equation of = 6 +. Its turning point form is: A = ( 6) + B = ( ) 6 C = ( ) + D = ( ) + REASONING Nikki wanted to keep a carnivorous plant, so after school she recorded the temperature on her windowsill for 8 hours ever da for several months. One summer evening the temperature followed the relationship t = (h ) +, where t is the temperature in degrees Celsius, h hours after pm. a Find the temperature on the windowsill at pm. b Show that the minimum temperature reached during the 8-hour period is C. c Find the number of hours it took for the windowsill to reach the minimum temperature. d Sketch a graph of the relationship between the temperature and the number of hours after recording began. Mark the turning point and the t-intercept on the graph. Topic 7 Quadratic functions 667

31 doc- int-776 A graph has an ais of smmetr at = and a -intercept of. If the graph s minimum value is, determine its equation in the form = a( h) + k. Demonstrate the correctness of our result. Students were asked to choose values for a, h and k and substitute them into the general equation = a( h) + k to form a quadratic equation. The graph shown was generated using the values., and for the three variables. Match the numerical values with the appropriate variable. Justif our answer. PrOblem SOlVIng 6 a Describe how to transform the parabola = ( ) to obtain the parabolas = ( ) and = ( ) +. b Another parabola is created b moving = ( ) so that its turning point is (, ). Write an equation for this parabola. 7 Find an equation for the parabola that has its turning point at (, ) and passes through the point (, ). 8 Find the -intercepts of = a( a) 9a in terms of a. 9 Use the three points (, ), (, ) and (, 7) to determine a, h and k in the equation = a( h) + k. Find the -intercepts of = a( h) k in terms of a, h and k. 7.8 Sketching parabolas of the form = ( + a)( + b) The equation = ( + a)( + b) form consists of a pair of linear factors ( + a) and ( + b) multiplied together. The -intercepts are found b setting each factor to, namel: ( + a) =, or = a ( + b) =, or = b. The -intercept can be found b letting = in the original equation. That is, = ( + a)( + b), or = ab. The ais of smmetr is half-wa between the -intercepts a and b. That is, a + b =. Substitute the -value of the ais of smmetr into = ( + a)( + b) to find the -coordinate of the turning point. WOrKed example Sketch the graph of = ( )( + ) b first finding the - and -intercepts and then the turning point. THInK WrITe/draW Write the equation. = ( )( + ) Find the -intercepts. -intercepts: when =, ( )( + ) = = or + = = = The -intercepts are and. (, ) 668 Maths Quest 9

32 Find the -intercept. -intercept: when =, = ( )( + ) = = 8 The -intercept is 8. Find the -value of the turning point At the turning point, = + b averaging the values of the two -intercepts. Find the -value of the turning point b substituting the -value of the turning point into the equation of the graph. When =, = ( ) ( + ) = = 9 6 State the coordinates of the turning point. The turning point is (, 9). 7 Sketch the graph. = ( )( + ) 8 (, 9) Quadratic equations written in the form = a + b + c can sometimes be factorised into intercept form. WOrKed example Sketch the graph of = THInK WrITe/draW Write the equation. = Factorise the epression on the righthand side of the equation, = ( + )( + ) Find the two -intercepts. -intercepts: when =, ( + )( + ) = + = = or + = = The -intercepts are and. Find the -intercept. -intercept: when =, = = 8 The -intercept is 8. + Find the -value of the turning point. At the turning point, = = = Topic 7 Quadratic functions 669

33 6 Find the -value of the turning point. When = = + + = = 7 State the turning point. The turning point is (, ). 8 Sketch the graph. = ( + )( + ) 8 reflection What are the advantages of having the equation of a parabola in intercept form? doc- 99 doc- doc- (, ) Eercise 7.8 Sketching parabolas of the form = ( + a)( + b) IndIVIdual PaTHWaYS PraCTISe Questions: 6,, COnSOlIdaTe Questions: a, b, f, b, e, h, k,,, 6, 9,,, Individual pathwa interactivit int-6 master Questions: a, b, d, c, f, i, l, FluenCY WE Sketch the graph of each of the following b first finding the - and -intercepts and then the turning point. a = ( + )( + 6) b = ( )( ) c = ( )( + ) d = ( + )( 6) e = ( )( + ) f = ( + )( ) WE Sketch the graph of each of the following. a = b = c = d = e = + 8 f = + g = h = 6 7 i = + j = 6 + k = + 6 l = Sketch the graph of a parabola whose -intercepts are at = and = 6 and whose -intercept is at =. State a possible equation of a graph with an ais of smmetr at = and one intercept at = Maths Quest 9

34 MC Which of the following quadratics has no -intercepts? A = + + B = + C = + + D = + + E None, all have -intercepts. UNDERSTANDING 6 Out on the cricket field, Michael Clarke chases the ball. He picks it up, runs metres towards the stumps (which are metres awa from where he picks up the ball), then throws the ball, which follows the path described b the quadratic equation =.( m)( n), where m and n are positive integers. The ball lands metre from the stumps, where another plaer quickl scoops it up and removes the bails. Taking the origin as the point where Michael picked up the ball initiall: a Find how far the ball has travelled horizontall while in flight. b Find the values of m and n. c Find the -intercepts. d Find the ais of smmetr. e Find the turning point. f Find the highest point reached b the cricket ball. g Find how far horizontall the ball has travelled when it reaches its highest point. h Sketch the flight of the ball, showing all of the relevant details on the graph. 7 McDonald s golden arches were designed b Jim Schindler in 96. The two arches both approimate the shape of a parabola. A large McDonald s sign stands on the roof of a shopping centre. The shape of one of the parabolas can be modelled b the quadratic function = + 8 7, where both and are measured in metres. The complete sign is supported b a beam underneath the arches. What is the minimum length required for this beam? 8 For the equation = ( )( 9): a determine the turning point b rewrite the equation in turning point form c epand both forms of the equation, showing that the are equivalent d sketch the equation showing the -intercepts and turning point. 9 The dimensions of a rectangular backard can be given b ( + ) m and ( ) m. Find the value of if the ard s area is 9 m. REASONING The height of an object, h(t), thrown into the air is determined b the formula h(t) = 8t + 8t, where t is time in seconds, and h is height in metres. a Will the object reach a maimum or a minimum? Eplain or show our reasoning. b What is this height and at what time? Daniel is in a car on a roller-coaster ride as shown in the following graph, where the height, h, is in metres above the ground and time for the ride, t, is in minutes. a The ride can be represented b three separate equations. Show that the first section of the ride is h = t from t = to t = 6. Find the other two equations for the ride. Topic 7 Quadratic functions 67

35 b What is the required domain for each section of the ride? h 6 (8.,.) (6, ) doc- 8 6 (, 9) PrOblem SOlVIng On a set of aes, sketch several parabolas of the form = + b. What do ou notice about the turning points of each graph sketched? Find a quadratic equation given the -intercepts of the parabola are and. Is there more than one equation possible? a The -intercepts of the quadratic equation = + are a and b. Find a and b. b Show that a = a + and b = b +. Find the equation of the parabola that passes through the points (, ), (, ) and (, ). 7.9 Applications Quadratic graphs and equations can be used to solve practical problems in science and engineering. Quadratic graphs and equations can be used to solve problems where a maimum or minimum needs to be found. When working with phsical phenomena, ensure that the solution achieved satisfies an phsical constraints of the problem. For eample, if measurements of length or time are involved, there can be no negative solutions. WOrKed example 6 A flare is fired from a acht in distress off the coast of Brisbane. The flare s height, h metres above the horizon t seconds after firing, is given b h = t + 8t +. a When will the flare fall into the ocean? b How high is the flare after seconds? c At which other time will the flare be at the same height? d For how long is the flare above the lowest visible height of 6 m? t 67 Maths Quest 9

36 THInK a Substitute h = into the equation and solve for t using the Null Factor Law. Since t cannot equal seconds, answer the question with the appropriate solution. b Substitute t = into the equation and solve for h. Answer the question. c Substitute h = 8 into the equation, rearrange, factorise and solve for t. d Substitute h = 6 into the equation, rearrange, factorise and solve for t. Answer the question. Eercise 7.9 Applications IndIVIdual PaTHWaYS PraCTISe Questions:, 9, COnSOlIdaTe Questions:, 6, 7,,, WrITe Individual pathwa interactivit int-6 a = t + 8t + = t 9t = t t + t = or t = seconds b h = + 8() + = = 8 The flare is 8 metres high after seconds. c 8 = t + 8t + = t + 8t 8 = t 9t + t 7 t t = 7 seconds d 6 = t + 8t + = t + 8t 6 = t 9t + 8 t 6 t t = seconds and 6 seconds The flare is above 6 m for seconds (between and 6 seconds). master Questions:, 7 FluenCY A spurt of water emerging from an outlet just below the surface of an ornamental fountain follows a parabolic path described b the equation h = + 8 where h is the height of the water and is the horizontal distance from the outlet in metres. a Sketch the graph of h = + 8. b State the maimum height of the water above the surface of the fountain. The position d metres below the starting point of a ball when it is dropped from a great height is given b the equation d =.9t. If it falls for second, it drops.9 m, giving a value of d =. m. a How far has it dropped after: i seconds? ii 7 seconds? b Sketch a graph of this relation. reflection What are some important things to remember when solving practical problems with quadratic graphs and equations? Topic 7 Quadratic functions 67

37 A car travels along a highwa for a number of minutes according to the relationship P = t + t, where P is the distance from home in metres and t is time in minutes. a What is the distance from home when t =? b How long does it take the car to reach home? UNDERSTANDING WE6 A rocket fired from Earth travels in a parabolic path. The equation for the path is h =.d + d, where h is the height in km above the surface of the earth and d is the horizontal distance travelled in km. a Find the height of the rocket after: i km ii 6 km. b How far awa does the rocket land? c What is the maimum height of the rocket and how far did it travel before it reached this height? d Sketch the path of the rocket. The height of a golf ball hit from the top of a hill is given b the quadratic rule h = t + t +, where h is in metres and t in seconds. a From what height was the golf ball hit? b What was the height of the ball after seconds? c When does the golf ball hit the ground? d What is the maimum height the ball reaches? e Sketch the graph of the flight of the ball. 6 Cave Ltd manufactures tedd bears. The dail profit, $P, is given b the rule P = n + 7n, where n is the number of tedd bears produced. a If the produce bears a da, what is the profit? b Sketch the graph of P for appropriate values of n. c How man bears do the need to produce before the start making a profit? d What is the maimum profit and how man tedd bears do the need to manufacture to make this amount? 7 A school plaground is to be mulched and the grounds keeper needs to mark out a rectangular perimeter of 8 m. m m m m 67 Maths Quest 9

38 a Find the perimeter of the plaground in terms of and. b Show that =. c Find the area to be mulched, A, in terms of onl. d Sketch the graph of A against for suitable values of. e Determine the maimum area of the plaground. f What are the dimensions of the plaground for this maimum area? 8 A woman wished to build a fence around part of her backard, as shown in the diagram. The fence will have one side abutting the wall of the house. She has enough fencing material for m of fence. m a Show that =. b Write an epression for the area enclosed b the fence in terms of alone. c Determine the dimensions of the fence such that the area is a maimum and calculate that area. REASONING 9 A basketball thrown from the edge of the court to the goal shooter is described b the formula h = t + 6t +, where h is the height of the basketball in metres after t seconds. a What was the height of the ball when it was first thrown? b What was the height of the basketball after s? c Show that the ball was first at a height of 6 m above the ground at second. d During which time interval was the ball above a height of 9 m? e Plot the path of the basketball. f What was the maimum height of the basketball during its flight? g How long was the ball in flight if it was caught at a height of m above the ground on its downward path? h If no one caught the ball, eplain when it would have landed on the ground. Jack and his dog were plaing outside. Jack was throwing a stick in the air for his dog to catch. The height of the stick (in metres) followed the equation h = t t. a The graph of h is a parabola. Is the parabola upright or inverted? b Factorise the epression on the right-hand side of the equation. m m House Topic 7 Quadratic functions 67

39 c Find the t-intercepts. These will be the two points at which the stick is on the ground, once at take-off and once at landing. How long does the stick remain in the air? (For simplicit, assume that Jack throws the stick from ground level.) d Justif whether the parabola has a maimum or minimum turning point, and find its coordinates. e What will be the maimum height reached b the stick? Eplain. f Use the information ou have found to produce a sketch of the path of the stick. An engineer wishes to build a footbridge in the shape of an inverted parabola across a -m wide river. The bridge will be smmetrical and the greatest difference between the lowest part and highest part will be m. Taking the origin as one side, show that the equation that models this footbridge is =.( ). Eplain what and are, and state the domain. PrOblem SOlVIng A car engine spark plug produces a spark of electricit. The size of the spark depends on how far apart the terminals are. The percentage performance, Z, of a certain brand is thought to be Z = (g.) +, where g is the distance between the terminals. a Sketch the graph of Z. b When is the performance greatest? c From our graph, find the values of g for which the percentage performance is greater than %. The monthl profit or loss, p (in thousands of dollars), for a new brand of soft drink is given b p = ( 7.) +., where is the number (integer) of months after its introduction (when = ). a In which month was the greatest profit made? b Between which months did the compan first make a profit? NASA uses a parabolic flight path to simulate zero gravit and the gravit eperienced on the moon. Use the internet to investigate the flight path and create a mathematical model to represent the flight path. An arch bridge is modelled in the shape of a parabolic arch. The arch span is m wide and the maimum height of the arch above water level is. m. A floating platform m wide is towed under the bridge. What is the greatest height of the deck above water level if the platform is to be towed under the bridge with at least a cm clearance on either side? CHa allenge 7. α 676 Maths Quest 9

40 number and algebra ONLINE ONLY 7. Review The Maths Quest Review is available in a customisable format for students to demonstrate their knowledge of this topic. The Review contains: Fluenc questions allowing students to demonstrate the skills the have developed to efficientl answer questions using the most appropriate methods Problem Solving questions allowing students to demonstrate their abilit to make smart choices, to model and investigate problems, and to communicate solutions effectivel. A summar of the ke points covered and a concept map summar of this topic are available as digital documents. Review questions Download the Review questions document from the links found in our ebookplus. Language int-69 int-78 applications ais of smmetr constant cross-section dilation domain horizontal translation intercept intercept form inverted maimum minimum modelled parabola parabolic shape quadratic function turning point vertical translation -intercept -intercept int- Link to assesson for questions to test our readiness FOr learning, our progress as ou learn and our levels OF achievement. assesson provides sets of questions for ever topic in our course, as well as giving instant feedback and worked solutions to help improve our mathematical skills. The stor of mathematics is an eclusive Jacaranda video series that eplores the histor of mathematics and how it helped shape the world we live in toda. Catapults and projectiles (eles-7) looks at how catapults were used in war for devastating effect for centuries. Different tpes of catapults are eplored, as well as how mathematics can determine the projectiles paths. Topic 7 Quadratic functions c7quadraticfunctions.indd /7/ :9 AM

41 number and algebra <InVeSTIgaTIOn> InVeSTIgaTIOn FOr rich TaSK Or <number and algebra> FOr PuZZle rich TaSK Constructing a parabola 678 Maths Quest 9 c7quadraticfunctions.indd 678 /7/ :9 AM

42 Forming a parabola b folding paper Take a sheet of A paper. Cut it into two pieces b dividing the longer side into two. Onl one of the halves is required for this investigation. Along one of the longer sides of our piece of paper, mark points that are equall spaced cm apart. Start with the fi rst point being on the ver edge of the paper. Turn over the piece of paper and mark a point, X, cm above the centre of the edge that has the markings on the reverse side. Fold the paper so that the fi rst point ou marked on the edge touches point X. Make a sharp crease and open the paper fl at. Fold the paper again so that the second mark touches the point X. Crease and unfold again. Repeat this process until all the marks have been folded to touch point X. With the paper fl at and the point X facing up, ou should notice the shape of a parabola appearing in the creases. Trace the curve with a pencil. The point X is called the focus of the parabola. Consider the parabola to represent a mirror. Ras of light from the focus would hit the mirror (parabola) and be reflected. The angle at which each ra hits the mirror is the same size as the angle at which it is reflected. Using our curve traced from our folding activit, accuratel draw a series of lines to represent ras of light from the point X to the parabola (mirror). Use a protractor to carefull measure the angle each line makes with the mirror and draw the path of these ras after refl ection in the mirror. Draw a diagram to describe our fi nding from question above. Provide a brief comment on our description. Retrace our parabola onto another sheet of paper. Take a point other than the focus and repeat the process of refl ection of ras of light from this point b the parabolic mirror. Draw a diagram to describe our fi nding from question above. Provide a brief comment on our description. 6 Give eamples where these sstems could be used in societ. Topic 7 Quadratic functions 679

43 <InVeSTIgaTIOn> number and algebra FOr rich TaSK Or <number and algebra> FOr PuZZle COde PuZZle Whatever happened to Australia s first -cent coin of 966? Match the letters for each of the features of the parabolas below with the correct answer from the list at the right to answer the code puzzle. = + + A = turning point = C = -intercepts = D = -intercept = = ( ) + L = -intercept = M = turning point = = ( + )( ) T = -intercepts = U = turning point = V = -intercept = = ( + )( 6) E = -intercepts = F = -intercept = G = turning point = = + N = smaller -intercept = O = turning point = = ( + ) W = turning point = Y = -intercept = = ( ) H = turning point = I = -intercept = = ( + )( + ) P = larger -intercept = R = turning point = S = -intercept = = 7 = (, ) (.,.) (, ) 6 = and = 6 7 (, ) 8 (, ) 9 = (, ) = (, ) = and = (, ) = and = 6 = 7 = 8 (.,.) 9 (, ) (, ) Maths Quest 9

44 Activities 7. Overview Video The stor of mathematics: Catapults and projectiles (eles-7) 7. graphs of quadratic functions digital doc SkillSHEET (doc-989): Equation of a vertical line Interactivit IP interactivit 7. (int-) Graphs of quadratic functions 7. Plotting points to graph quadratic functions digital docs SkillSHEET (doc-99): Substitution into quadratic equations SkillSHEET (doc-99): Plotting coordinate points Interactivit IP interactivit 7. (int-) Plotting points to graph quadratic functions 7. Sketching parabolas of the form = a Interactivities = a (int-79) IP interactivit 7. (int-6) Sketching parabolas of the form = a 7. Sketching parabolas of the form = a + c Interactivities Vertical translation: = + c (int-9) IP interactivit 7. (int-7) Sketching parabolas of the form = a + c digital doc WorkSHEET 7. (doc-9): Quadratic functions I 7.6 Sketching parabolas of the form = ( h) Interactivities Horizontal translation: = ( h) (int-9) IP interactivit 7.6 (int-8) Sketching parabolas of the form = ( h) To access ebookplus activities, log on to 7.7 Sketching parabolas of the form = ( h) + k Interactivities Parabolas of the form = ( h) + k (int-8) IP interactivit 7.7 (int-9) Sketching parabolas of the form = ( h) + k digital doc WorkSHEET 7. (doc-): Quadratic functions II 7.8 Sketching parabolas of the form = ( + a)( + b) digital docs SkillSHEET (doc-99): Solving quadratic equations of the form ( + a)( + b) = SkillSHEET (doc-): Factorising quadratic trinomials of the form a + b + c where a = SkillSHEET (doc-): Solving quadratic trinomials of the tpe a + b + c = where a = WorkSHEET 7. (doc-): Quadratic functions III Interactivities Sketching parabolas (int-776) IP interactivit 7.8 (int-6) Sketching parabolas of the form = ( + a)( + b) 7.9 applications Interactivit IP interactivit 7.9 (int-6) Applications 7. review Interactivities Word search (int-69) Crossword (int-78) Sudoku (int-) digital docs Topic summar (doc-86) Concept map (doc-87) Topic 7 Quadratic functions 68