Mth 4108-1 c (ans)
- 1. MTH-4108 C Quadratic Functions ANSWER KEY 1. Graph the following equations. Be sure to include the coordinates of at least 5 points, including the vertex, the zeros (if any) and y-intercept. a) y = 3.2x2 + 4x y −b −∆ , 2a 4 a Vertex: − 4 − 16 , 6.4 12.8 ( − 0.625,−1.25) x −b± ∆ 2a Zeros: − 4 ± 16 6.4 { − 1.25,0} y = 3.2(0) + 4(0) y-intercept: y=0 b) y = ¼x2 – 3x +1 y Vertex: −b −∆ , 2a 4a 3 − ( 9 − 4( 0.25)(1) ) , 0.5 1 ( 6,−8) x −b± ∆ 2a Zeros: 3 ± 8 0.5 { 0.343,11.65} y-intercept: y = 0.25(0) − 3(0) + 1 y =1
- 2. c) y = –3x2 + x – 2 Vertex: y −b −∆ , 2a 4a − 1 − (1 − 4( − 3)( − 2 ) ) , −6 − 12 ( 0.166,−1.9166) −b± ∆ 2a x Zeros: − 1 ± − 23 −6 { ∅} y-intercept: y = −3(0) + (0) − 2 y = −2 /30 2. Solve the following equations by factoring: a) 8x2 – 2x = 0 2 x(4 x − 1) = 0 2x ⇒ 2x = 0 ⇒ x = 0 ( 4 x − 1) ⇒ 4 x = 1 ⇒ x = 1 4 b) 3/5x2 – 2/5x + 1/5= 0 ( ) 1 3x 2 − 2 x − 1 = 0 5 3x 2 − 3x + 1x − 1 = 0 3x( x − 1) + 1( x − 1) (3x + 1)( x − 1) = 0 ( 3x + 1) ⇒ 3x = −1 ⇒ x = − 13 ( x − 1) ⇒ x = 1 /5 3. Identify the true statement(s) below which pertain to quadratic functions in the form: y = ax2 + bx + c a) If a = 0, it means that the function is no longer quadratic. TRUE ____________ FALSE b) If c = 0, it means that the vertex is at (0,0). ____________ TRUE c) If b = 0, it means that the function is situated on the y-axis. ____________ d) If a > 0, it means that the function opens upward. TRUE ____________ e) If c > 0, it means that the function is above the x-axis. FALSE ____________ /5
- 3. 4. Solve the following equations using the quadratic formula. Clearly indicate the value of ∆ and round your answers to the nearest thousandth when necessary. 5 a) /3x2 – 2/5x + 1/2= 0 Zeros: −b± ∆ 2a 0.4 ± ( 0.16 − 4(1.667 )( 0.5) ) 1 0.4 ± − 3.1733 1 { ∅} b) 2x – 3x2 +2 = 0 −b± ∆ 2a Zeros: − 2 ± ( 4 − 4( − 3)( 2 ) ) −6 { − 0.5485,1.2152} /10 5. Every month, Irene sells 6 dozen roses that she grows in her garden for $20 per dozen. For every additional dozen roses she grows, she can reduce her price by $2 per dozen. How many dozen roses should she grow every month in order to maximize her total sales? Complete the following table and write the equation that can be used to solve the problem. Your equation should be in the form y = ax2 + bx + c Number of Total number of Selling Price ($) Total Sales ($) additional dozens of roses y dozens of roses x 0 6 20 6 × 20 = 120 1 6+1=7 20 – 2 = 18 7 × 18 = 126 2 6+2=8 20 – 2(2) = 16 8 × 16 = 128 3 6+3=9 20 – 2(3) = 14 9 × 14 = 126 x 6+x 20 – 2x (20 – 2x)( 6 + x) Write the equation in the form ax2 + bx + c which illustrates this situation. ( 20 − 2 x )( 6 + x ) 120 + 20 x − 12 x − 2 x 2 y = −2 x 2 + 8 x + 120 /10
- 4. 6. Answer the following questions using the graph below: y x True or False? TRUE a) Point (-2,0) is a zero. ____________ TRUE b) Point (0,-1) is a vertex. ____________ FALSE c) The equation for the axis of symmetry is: y = 0. ____________ TRUE d) Point (-0,-1) is a minimum. ____________ FALSE e) Point (2,0) is the y-intercept. ____________ /5 7. Without calculating, write the equation in the form ax2 + bx + c which illustrates the following situation: A mother’s present age is triple her son’s age. In 15 years, the product of their ages will be 60 times her son’s age then. Let x = Son’s age Let 3x = Mother’s age ( x + 15)( 3x + 15) = 60( x + 15) 3 x 2 + 15 x + 45 x + 225 = 60 x + 900 3 x 2 − 675 = 0 /5
- 5. 8. The hypotenuse of a right triangle is 10 cm. Find the length of the other two sides if one side is 2 cm longer than the other, using the Pythagorean Theorem. Let x = small side Let 2 + x = LARGE side x 2 + ( x + 2) 2 = 10 2 x 2 + 2 x − 48 = 0 x 2 + x 2 + 4 x + 4 = 100 Zeros: ( x + 8)( x − 6) = 0 2 x 2 + 4 x − 96 = 0 { − 8,6} x 2 + 2 x − 48 = 0 2 + (6) = 8 ANS: The two sides measure 6 cm and 8 cm /5 9. A School principal paid $180 for a certain number of desks. If the desks cost 3 dollars less, she could have bought 5 more desks for the same price. How many desks did the principal order? Let x = number of desks ordered 180 180 −3= x x+5 180 − 3 x 180 = x 2 + 5 x − 300 = 0 x x+5 180 x + 900 − 3 x 2 − 15 x = 180 x Zeros: ( x − 15)( x + 20) − 3 x 2 − 15 x + 900 = 0 { − 20,15} x 2 + 5 x − 300 = 0 ANS: The principal ordered 15 desks. /10 10. The function h = 24t – 6t2 represents the height (h) in centimeters reached by a champion jumping frog after t seconds. Use the method of your choice to find the frog’s maximum height. Round your answers to the nearest hundredth. − b − ∆ − 24 − ( 576 ) Vertex: , ⇒ , ⇒ ( 2,24 ) 2a 4a − 12 − 24 /5
- 6. 11. Tarzan swings on a vine from a tall tree, across a river, and rises upwards to another tree in a parabolic arc. The equation f = 0.04x2 – 5x + 28 describes his motion, withboth the fall (f) and distance (x) between the two trees measured in feet. Use the method of your choice to answer the following questions: a) What is the distance between the two trees? −b± ∆ 5 ± (−5) 2 − 4( 0.04)( 28) 5 ± 20.52 ⇒ ⇒ 2a 2(0.04) 0.08 Zeros: { 5.876,119.123} 119.123 − 5.876 = 113.2475 b) What is the measure of the vertical drop? − b − ∆ 5 − ( 20.52 ) Vertex: , ⇒ , ⇒ ( 62.5,−128.25) 2a 4a 0.08 4(0.04) /10