Introduction to Graphing Quadratics
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1 Graphing Quadratic Functions Table of Contents 1. Introduction to Graphing Quadratics (19.1) 2. Graphing in Vertex Form Using Transformations (19.2) 3. Graphing in Standard Form (19.3) 4. Graphing in Factored Form (20.1 and 20.2) Introduction to Graphing Quadratics Quadratics Definition: Equations and expressions involving polynomials where the highest power is 2. Graphically: always a U shape Algebraically: Values increase at an increasing rate Also named parabola Parts of Quadratic Graph Vertex: Minimum vs. Maximum The botto m (or top) of U is calle d v erte x, o r tu rn i ng point. Th e vert ex of a para bola open ing upw ard is also calle d mini mum point. The vert ex of a para bola open ing dow nwa rd is also calle d maxi mum point. The x- in ter cept s are calle d r oots, or zero s. To find x - in ter cept s, set a x 2 + bx+ c = 0. The ends of grap h co ntinu e to posit iv e in fin ity (or nega tiv e in fin ity ) unle ss dom ain (the x's to be grap hed) is othe rwis e spec ified. 1
2 Sketch the Graph Sketch the Graph g(x)= 3x 2 X Y Domain: Range: Write a Function Given a Graph Write a Function Given a Graph 1. Use form g(x) = ax 2 2. Plug in x cord for x and y cord of g(x) and solve for a Satellite dishes reflect radio waves onto a collector by using a reflector dish shaped like a parabola. The graph shows the height h in feet of the reflector relative to the distance x in feet from the center of the satellite dish. Find the equation of the quadratic and describe what the function represents. Equation: Describe: Vertex? Point: (60, 12)? 2
3 Graphing in Vertex Form Using Transformations 3 Forms of Quadratic Equations Vertex Form: y= a(x h) 2 + k Standard Form: y = ax 2 + bx + c Factored Form: y = k(x a)(x b) Graphing in Vertex Form 1. Identify the vertex. 2. Plot vertex and draw axis of symmetry. 3. Create table of values (pick 2 x values bigger than vertex and 2 x values smaller than vertex.) Graph: y = (x + 3) 2 1 Identify the Vertex: End Behavior? Graph: y = (x 2) Identify the Vertex: End Behavior? X Y X Y 3
4 Up or Down?! Skinny or Fat? Transformations of Vertex Form Type of Transformation Details Graph: y = 2(x + 3) 2 1 a h k Vertical Transformation? Horizontal Transformation? Graph: y = (x 4)2 + 2 Vertical Transformation? Horizontal Transformation? 4
5 Graphing Using Standard Form Standard Form of Quadratics Standard Form: y = ax 2 + bx + c Standard Form: Find Vertex 1. Identify a, b and c. 2. Find x coordinate using formula: 3. Plug in x cord into original equation and solve for y coordinate. Conditions: a, b and c must be real numbers and not be zero. Let s Practice 1. Calculate the vertex. y = x 2 8x 15 a: b: c: x cord: Let s Practice 2. Calculate the vertex. y = x 2 4x +5 a: b: c: x cord: y cord: y cord: 5
6 Find Zeros/Roots 1. Factor original equation. 2. Set factors equal to zero and solve. 3. The two solutions are the x intercepts. Let s Practice 1. Find zeros/roots. y = x 2 + 8x + 15 Factor: Solve: x intercepts: Graph: y = x 2 + 5x + 4 Identify a: b: c: Graph: 6x + 8 = x 2 Identify a: b: c: Zeros: Zeros: The equation for motion of a projectile fired straight up at an initial velocity of 64 ft/s is 64 16, where h is height in feet and t is time in seconds. Find the time the projectile needs to reach its highest point. How high will it go? Graph: 2x 2 5 = 3 Identify a: b: c: Zeros: 6
7 A baseball coach used a pitching machine to simulate pop flies during practice. The quadratic function h(t) = 16t t + 5 models the height in feet of the baseball after t seconds. The ball leave the pitching machine and is caught at a height of 5 feet. How long is the baseball in the air? Graph: 3x 2 9 = 6 Identify a: b: c: Zeros: Graphing in Factored Form Graph: Factored Form Factored Form: y = k(x a)(x b) 1. Set factors (parentheses) equal to zero and solve. 2. Plot on graph. 3. X value of vertex is half way between zeros (from step 1). 4. Plug in x value to equation to y value of vertex. Graph: y = (x 1)(x 3) Zeros: A tennis ball is tossed upward from a balcony. The height of the ball in feet can be modeled by the function y = 4(2x + 1)(2x 3) where x is the time in seconds after the ball is released. Find the maximum height of the ball and the time it takes the ball to reach this height. Determine how long it takes the ball to hit the ground. 7
8 Graph: y = 2(x + 4)(x + 2) Graph: y = x 2 4x 5 Zeros: Zeros: Graph: 6x + 8 = x 2 Mini Quiz Zeros: 8
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