Review Topic 3: Quadratic relationships

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1 Review Topic 3: Quadratic relationships Short answer Solve for x. a ( x + 4) - 7( x + 4) - 8 = x = 3 x( x- ) + b c x = x - - d 3+ x = x Solve the following quadratic inequations. a x - 5x- 3> b - x ³ c x + x+ 5 ³ 3 Sketch the graphs of the following, showing all key points. a y= ( x- 3)( x+ ) b y= - ( x+ ) c d y x x = y³ - + x 4 4 Factorise over R. a - x + x+ 4 b 4x - x- 9 5 For what values of k does the equation kx - 4( x k + ) + 36= have no real roots? 6 a Use an algebraic method to find the coordinates of the points of intersection of the parabola y= x + x and the line y = x+. b Sketch c = + and y = x+ on the same set of axes. y x x Give an algebraic description of the region enclosed by the parabola the line y = x+. y x x = + and d i For what value of k is the line y = x + k a tangent to the parabola y= x + x? ii Sketch this tangent on the diagram drawn in part b, identifying the point of contact with the parabola. John Wiley & Sons Australia, Ltd

2 Multiple choice The solutions of the equation ( x- )( x+ ) = 4 are: A x=, x=- B x= 6, x=- C x=- 6, x= D x= 3, x=- E x=- 3, x= The values of x for which - 5x + 8x+ 3= A -.6, - B.6, - C -.3,.9 D.3, -.9 E -., - 3 are closest to: 3 For the graph of the parabola is correct? y= ax + bx+ c shown, with D = b - 4ac, which statement A a > and D > B a > and D < C a < and D < D a < and D > E a > and D = 4 The parabola with equation y = x The equation of the image is: A y= ( x- 4) + 3 B y= ( x- 3) + 4 C y= ( x+ 4) + 3 D y= ( x+ 3) - 4 E y=- 4x + 3 is translated so that its image has its vertex at ( - 4,3). 5 If x + 4x- 6 is expressed in the form ( x+ b) + c then the values of b A b=, c=- B b=-, c=- C b= 4, c=- D b=- 4, c=- E b=, c=- 8 and c would be: John Wiley & Sons Australia, Ltd

3 6 The equation of the parabola shown is: A B C D E y x x = y= x + x-.5 y= x - x- 4 y=.5x - x- y= + - ( x ) 7 The solution set of { x: x < 4 x} is: A { x: x< 4} B { x: - 4< x< } C { x:< x< 4} D { x: x< } È { x: x> 4} E { x: x<- 4} È { x: x> } 8 A quadratic graph touches the x-axis at x =- 6 and cuts the y-axis at y =-. Its equation is: A y= ( x+ 6) - B y = ( x+ 6)( x+ ) 5 C y = x D y = ( x+ 6) 8 5 E y=- ( x+ 6) 8 9 The x-coordinates of the points of intersection of the parabola line x- y = can be determined from the equation: A B C D E 3x - x+ = 3x - x+ 3= x - 6x+ = 3x - 8x+ = (x - ) = y= x - x+ 3 with the John Wiley & Sons Australia, Ltd 3

4 The maximum value of A 5 B 4 C 3 D E - 4- x- x is: John Wiley & Sons Australia, Ltd 4

5 Extended response At a winter skiing championship, two competitors, one from Japan and the other from Canada, compete for the gold medal in one of the jump events. Each competitor leaves the ski run at point S and travels through the air, landing back on the ground at some point G. The winner will be the competitor who covers the greater horizontal distance OG. The Canadian skier jumps first and her height ( h metres) above the ground is described by h=- ( x - 6 x- 7), where x metres is the horizontal distance travelled. 35 a Show that the point S is metres above O. b How far does the Canadian skier jump? The Japanese skier jumps next and she reaches a maximum height of 35 metres above the ground after a horizontal distance of 3 metres has been covered. c Assuming the path is a parabola, form the equation for h in terms of x which describes this competitor s path. d Decide which competitor receives the gold medal. Your decision should be supported with appropriate mathematical reasoning. The diagram shows the arch of a bridge where the shape of the curve, OAB, is a parabola. OB is the horizontal road level. Taking O as the origin, the equation of the curve OAB is y=.5x-.35x. All measurements are in metres. a Calculate the length of OB, the span of the bridge. b How far above the road is the point A, the highest point on the curve? c A car towing a caravan needs to drive under the bridge. The caravan is 5 metres wide and has a height of metres. Only one single lane of traffic can pass under the bridge. Explain clearly, using mathematical analysis, whether the caravan can be towed under this bridge. To avoid accidents, the bridge engineers decide to place height and width limits. Only vehicles whose height and width fit into the greatest allowable dimensions are permitted to travel under the bridge. John Wiley & Sons Australia, Ltd 5

6 P ( xy, ) lies on the curve and is a corner of the rectangle formed by the height and width restrictions. d Express the width w of the rectangle in terms of x. e If the height restriction is 3. m, calculate the x-coordinate of P. f Would the caravan be permitted to be towed under the bridge under these restrictions? 3 ABCD is a rectangle of length one more unit than its width. Point F lies on AB and divides AB in the ratio x : so that AF is x units in length and FB is unit in length. Point G lies on DC and divides DC in the same ratio, x :. a Draw a diagram showing this information. b What is the width of rectangle ABCD? c If the area of the square AFGD is one more square unit than the area of the rectangle FBCG, show that x - x- =. d Hence find the value of x in simplest surd form. e The value found for x is called the Golden ratio and usually given the symbol f. Calculate f and give its relationship to the other root of the equation x - x- =. f Show f f = - and explain this relationship using the equation x - x- =. 4 Ignoring air resistance, the path of a cricket ball hit by a batsman can be considered to travel on a parabolic path which starts at the point (,) where the ball is struck by the batsman. Let x metres measure the horizontal distance of the ball from the batsman in the direction the ball travels, and y metres measure the vertical height above the ground that the ball reaches. John Wiley & Sons Australia, Ltd 6

7 A batsman hits a cricket ball towards a fielder who is 65 metres away. The ball is struck with a horizontal speed of 8 m/s, which is assumed to remain constant throughout the flight of the ball. On its way, the ball reaches a maximum height of 4.9 metres after second. a Calculate the coordinates of the turning point of the quadratic path of the ball. b Form the equation of the path of the ball. The fielder starts running forward at the instant the ball is hit and catches it at a height of.3 metres above the ground. c Calculate the time it takes the fielder to reach the ball. d Hence obtain the uniform speed in m/s with which the fielder runs in order to catch the ball. John Wiley & Sons Australia, Ltd 7

8 Review answers Short answer a x =± b x = 3± c x =± 4 d 9 x = 4 a x<- or x> 3 b - x c xî R 3 a x-intercepts (3,),( -,) ; y-intercept (, - 6) ; turning point (, - 8) b x-intercepts ( 3, ), (, ); y-intercept (, 3); turning point ( -,) c No x-intercepts; y-intercept (, 9); turning point ( -.5,8.75) John Wiley & Sons Australia, Ltd 8

9 d x-intercepts ( ±,) ; y-intercept and turning point (,4); region above closed 4 a - ( x- - 3)( x- + 3) b 4 æ + x ö æ - x ö ç ç 4 è ø è ø 5 < k < 4 6 a ( -,),(,3) b Parabola: x-intercepts ( -,),(,) ; turning point ( -, - ) Line through (, ), (, ). Both graphs meet at ( -,),(,3). c y x+ and d i k =- 4 y³ x + x ii The tangent is added to the graph in part b. Point of contact: æ 3ö ç -, - è 4ø. Multiple choice D C 3 B 4 C 5 A 6 B 7 C 8 E 9 B A John Wiley & Sons Australia, Ltd 9

10 Extended response a h=- ( x - 6 x- 7) 35 When x =, h =- ( - 7) 35 \ h = and the point S is (,). Therefore, S is metres above O. b 7 metres c h=- ( x- 3) d Japanese competitor wins a 8 metres b 5 metres c The caravan can be towed under the bridge. d x - 8 e 6.4 f No 3 a b c d e f x units Area measure of rectangle AFGD is x. Area measure of rectangle FBCG is x= x. \ x = x+ \ x - x- = =- f + 5 f - = - and: + 5- = = 5- John Wiley & Sons Australia, Ltd

11 - 5 =- f 5- = = f - As x = f is a root of f - f - = f f ( - ) = f - = f 4 a (8,4.9) x - x- =, y=- ( x- 8) seconds 7 m/s b c d John Wiley & Sons Australia, Ltd