Graphs and Solutions for Quadratic Equations

Transcription

1 Format y = a + b + c where a 0 Graphs and Solutions for Quadratic Equations Graphing a quadratic equation creates a parabola. If a is positive, the parabola opens up or is called a smiley face. If a is negative, the parabola opens down or is called a frown. All parabolas have a verte (the highest or lowest point on the parabola). To calculate the coordinate of the verte ( v ): v = - b To calculate the y coordinate of the verte (y v ), plug v into the original function: y v = a( v ) + b( v ) + c To find the zeros (the values of for which y = 0 and where the parabola crosses the -ais, factor and use the Principle of Zero Products OR apply the Quadratic Formula. 1

2 Eample William Tell Bow and Arrow Problem (Beginning Algebra, Hunter et. al., pg. 75) William Tell shoots an arrow straight up with an initial velocity of 160 feet per second. The height (in feet) of the arrow is given by the equation h = -16t t Question E: When will arrow reach ma height? Ma height corresponds to the h (or y) coordinate of the verte and when corresponds to the t (or ) coordinate of the verte. t t v v - b = - b = = = 5 t v = 5 seconds h v = -16(5) + 160(5) = 400 feet The coordinates of the verte are (5,400) Question F: When will arrow hit ground? Arrow hitting ground corresponds to h = 0 (or y = 0) and when corresponds to t (or ). In other words, this question is asking us to find the zeros for this quadratic equation.

3 Method 1 for Finding Zeros: Factor and Apply Principle of Zero Products h = -16t t Start with original equation 0 = -16t t Replace h with 0 0 = -16t (t -10) Factor (opposite of distribution) -16t = 0 and/or t - 10 = 0 Principle of Zero Products If (a)(b) = 0, then a = 0 and/or b = 0 t = 0 and/or t = 10 The arrow will hit the ground at 0 seconds (before flight) and at 10 seconds. The coordinates are (0,0) and 10,0). Solve for variable Conclusion Method for Finding Zeros: Use Quadratic Formula h = -16t t 0 = -16t t a = -16 b = 160 c = 0 b b 4ac 160 (160) ( 16) 4( 16)(0) 3

4 and/or 3 = 0 and/or = 10 Conclusion: y = 0 when = 0 and = 10, so h = 0 (arrow hits the ground) when t = 0 seconds and t = 10 seconds. The coordinates of these points are (0,0) and (10,0). Using the Quadratic Formula to Find X for a Non- Zero Value of Y If we want to use the quadratic formula to determine when William Tell s arrow is 56 feet in the air: Replace h (the y variable) with 56 in the original equation. Use algebra to set the equation equal to zero. Identify values of a, b, and c and plug into quadratic formula. Calculate corresponding t values (the variable). Remember, these t values are for h = 56 feet. 4

5 h = -16t t 56 = -16t t 0 = -16t t 56 a = -16 b = 160 c = -56 b b 4ac = - ± (160) 4( 16)( 56) ( - 16) = - ± = ± 96 = and/or = = and/or = 8 Conclusion: The arrow is at a height of 56 feet at seconds and 8 seconds. The coordinates of these points are (,56) and (8,56). 5

6 Graphing With coordinates of the verte, coordinates of the zeros, and coordinates for the additional points, we can plot these points and sketch the parabola. t h The verte. Formula for -coord. of verte used to calculate t. Then t plugged into quad. equation to calculated h. 0 0 The zeros : points where h = 0 and 10 0 parabola intercepts t-ais. Quad. Formula used to calculate t-values. 56 The points for which h = 56. Quad Formula used to calculate t-values. William Tell Problem (Pg. 75) Height of Arrow (Feet) Time (Seconds) 6