Bellwork: Describe each transformation: f(x) = -(x – 1)2 + 4 Algebra II

Graphing Quadratic Functions Algebra II

Three Forms of a Quadratic 1. Vertex Form: y = a(x – h)2 + k 2. Standard Form: y = ax2 + bx + c 3. Intercept Form: y = a(x – p)(x – q) Algebra II

Vocabulary: Axis of symmetry: the line that splits a parabola through the vertex Minimum value: the vertex of a parabola that opens up Maximum value: the vertex of a parabola that opens down Algebra II

Vertex: Min/Max values Algebra II

Graphing from vertex form y = a(x – h)2 + k Vertex: (opposite of h, same as k) Axis of symmetry (AOS): x = opposite of h. The parabola opens up if a > 0 and opens down if a < 0 The parabola is: wider that x2 if |a| < 1 narrower if |a| > 1 same width if |a| = 1. Algebra II

Example 1 Graph: y = - ½(x + 3)2 + 4 Opens down Wider than x2 Vertex: (-3,4) AOS: x = -3 Table  Reflect x -2 -1 y 3.5 2 Algebra II

Example 2 Graph: y = 2(x – 1)2 + 3 Opens up Narrower than x2 Vertex: (1,3) AOS: x = 1 Table  Reflect x 2 3 y 5 11 Algebra II

Example 3 Graph: y = -(x + 5)2 + 2 Opens down Same width as x2 Vertex: (-5,2) AOS: x = -5 Table  Reflect x -4 -3 y 1 -2 Algebra II

Example 4 Graph: y = 2(x – 1)2 + 1 Opens up Narrower than x2 Vertex: (1,1) AOS: x = 1 Table  Reflect x 2 3 y 3 9 Algebra II

Example 5 Graph: y = -2(x + 2)2 Opens down Narrower than x2 Vertex: (-2,0) AOS: x = -2 Table  Reflect x -1 y -2 -8 Algebra II

Graphing from standard form f(x) = ax2 + bx + c AOS: Vertex: The parabola opens up when a>0 and opens down when a < 0. The graph is narrower than the graph of f(x) = x2 when |a| > 1 and wider when |a| <1. The y-intercept is c. So the point (0,c) is on the parabola. Algebra II

Graphing in Standard Form Algebra II

Example 6 Graph: y = x2 – 2x – 5 AOS: x = - b / (2a) x = 2 / (21) Opens up, Same width as x2 AOS: x = - b / (2a) x = 2 / (21) x = 1 y = (1)2 – 2(1) – 5 Vertex: (1, -6) Table  Reflect x 2 3 y -5 -2 Algebra II

Opens up, Narrower than x2 Example 7 Graph: y = 2x2 – 4x + 3 Opens up, Narrower than x2 AOS: x = - b / (2a) x = 4 / (22) x = 1 y = 2(1)2 – 4(1) + 3 Vertex: (1, 1) Table  Reflect x 2 3 y 3 9 Algebra II

Example 8 Graph: y = ½x2 + x – 6 AOS: x = - b / (2a) x = -1 / (2½) Opens up, Wider than x2 AOS: x = - b / (2a) x = -1 / (2½) x = -1 y = ½(-1)2 + (-1) – 6 Vertex: (-1, -6.5) Table  Reflect x 1 y -6 -4.5 Algebra II

Opens down, Narrower than x2 Example 9 Graph: y = -2x2 + 1 Opens down, Narrower than x2 AOS: x = - b / (2a) x = -0 / (2-2) x = 0 y = -2(0)2 + 1 Vertex: (0, 1) Table  Reflect x 1 2 y -1 -7 Algebra II

Opens up, Narrower than x2 Example 10 Graph: y = 2x2 – 4x – 1 Opens up, Narrower than x2 AOS: x = - b / (2a) x = 4 / (22) x = 1 y = 2(1)2 – 4(1) – 1 Vertex: (1, -3) Table  Reflect x 2 3 y -1 5 Algebra II

Graphing from intercept form y = a(x – p)(x – q) x-intercepts: (opposite of p, 0) , (opposite of q, 0) AOS: x = (opp p + opp q) (half way between) 2 Vertex: (opp p + opp q , f(opp p + opp q) ) 2 2 Opens up if a > 1 and down if a < 1 ***Do not have to make a table*** Algebra II

Opens down, Same width as x2 Example 11 Graph: y = -(x + 2)(x – 4) Opens down, Same width as x2 Intercepts: (-2, 0), (4, 0) AOS: x = (- p + - q) / 2 x = (-2 + 4) / 2 x = 1 y = -(1 + 2)(1 – 4) Vertex: (1, 9) Algebra II

Example 12 Graph: y = ½(x – 6)(x – 4) Intercepts: (6, 0), (4, 0) Opens up, Wider than x2 Intercepts: (6, 0), (4, 0) AOS: x = (- p + - q) / 2 x = (6 + 4) / 2 x = 5 y = ½(5 – 6)(5 – 4) Vertex: (5, -½) Algebra II

Example 13 Graph: y = -½x(x – 5) Intercepts: (0, 0), (5, 0) Opens down, Wider than x2 Intercepts: (0, 0), (5, 0) AOS: x = (- p + - q) / 2 x = (0 + 5) / 2 x = 2.5 y = -½(2.5)(2.5 – 5) Vertex: (2.5, 3.125) Algebra II

Opens down, Narrower than x2 Example 14 Graph: y = -2(x – 3)(x + 1) Opens down, Narrower than x2 Intercepts: (3, 0), (-1, 0) AOS: x = (- p + - q) / 2 x = (3 + -1) / 2 x = 1 y = -2(1 – 3)(1 + 1) Vertex: (1, 8) Algebra II

Example 15 Graph: y = ⅓(x – 2)(x + 4) Intercepts: (2, 0), (-4, 0) Opens up, Wider than x2 Intercepts: (2, 0), (-4, 0) AOS: x = (- p + - q) / 2 x = (2 + -4) / 2 x = -1 y = ⅓(-1 – 2)(-1 + 4) Vertex: (-1, -3) Algebra II

Review: In which form is each equation written: Intercept, Standard, or Vertex ? y = 2(x – 3)2 y = 3(x – 2)(x + 1) y = ½x2 + 3 y = ½x(x – 2) y = –(x – 3)2 + 2 y = 2x2 + 3x y = 3x(x + 2) y = 2x2 – 5 y = -2(x + 1)(x + 3) y = 2(x – 1)2 vertex intercept standard Algebra II

Graphing Quadratic Functions 2.2 B Graphing Quadratic Functions Algebra II

Vertex: AOS: Domain: Range: Increasing: Decreasing: Graph the function. Label the vertex and axis of symmetry. Describe the domain and range and where the graph is increasing and decreasing. Vertex: AOS: Domain: Range: Increasing: Decreasing: Algebra II

Vertex: AOS: Domain: Range: Increasing: Decreasing: Graph the function. Label the vertex and axis of symmetry. Describe the domain and range and where the graph is increasing and decreasing Vertex: AOS: Domain: Range: Increasing: Decreasing: Algebra II

Vertex: AOS: Domain: Range: Increasing: Decreasing: Graph the function. Label the vertex and axis of symmetry. Describe the domain and range and where the graph is increasing and decreasing. Vertex: AOS: Domain: Range: Increasing: Decreasing: Algebra II

Graph the function. Label the x intercepts, vertex, and axis of symmetry. Algebra II

Word Problems A softball player hits a ball whose path is modeled by f(x) = -.0.0005x2 + 0.2127x + 3, where x is the distance from home plate (in feet) and y is the height of the ball above the ground (in feet). What is the highest point this ball will reach? If the ball was hit to center field which has an 8 foot fence located 410 feet from home plate, was this hit a home run? Explain. Algebra II

Word Problems The parabola shows the path of your first golf shot, where x is the horizontal distance (in yards) and y is the corresponding height (in yards). The path of your second shot can be modeled by the function f(x) = -0.02x(x – 80). Which shot travels farther before hitting the ground? Which travels higher? Algebra II

Word Problems The equation for the percent of test subjects that felt comfortable at a given temperature x is y = –3.678x2 + 527.3x – 18,807. What temperature made the greatest percent of test subjects comfortable? At that temperature, what percent of people felt comfortable?

Example 8 The Golden Gate Bridge in San Francisco has two towers that rise 500 feet above the road and are connected by cables as shown. Each cable forms a parabola with the equation y = 1/8960(x – 2100)2 + 8. What is the distance between the two towers? What is the height of the cable above the road at its lowest point?

Example 9 The archway that forms the ceiling of a tunnel can be modeled by the equation y = –0.0355x2 + .923x + 10 where x is the horizontal distance in feet and y is the height in feet from the ceiling to the floor. How many feet from the walls does the ceiling reach its maximum height? What is the maximum height?

Example 10 The equation for the jump of a cricket is y = -¾x(x – 2). What is the maximum height of the cricket and how far can the cricket jump?

Word Problems: Although a stadium field of synthetic turf appears to be flat, its surface is actually shaped like a parabola. This is so that rainwater runs off to the sides. If we take a cross section of the turf, the surface can be modeled by, y = -0.000234(x – 80)2 + 1.5, where x is the distance from the left end of the field and y is the height of the field. What is the width of the field? How high is the field? Algebra II

Word Problems: You maintain a music-oriented web-site that allows subscribers to download audio and video clips. When the subscription price is $16 per year, you get 30,000 subscribers. For each $1 increase in price, you expect to lose 1,000 subscribers. How much should you charge to maximize your annual revenue? What is your maximum revenue? Algebra II

Word Problems The owner of a gym charges $34 per month and has 48 members. For every $1 decrease in price they would gain four new members. What should the gym charge to maximize your annual revenue, and what is the maximum revenue? Algebra II

Word Problems: The height, h, in feet of an object above the ground is given by h = -16t2 + 64t + 190, t ≥ 0 where t is the time in seconds. Find the maximum height. How long does it take to get to that height? How high was the object when it was launched? Algebra II

Word Problems: The number of horsepower needed to overcome a wind drag on a certain automobile is given by N(s) = 0.005s2 + 0.007s − 0.031, where s is the speed of the car in miles per hour. How much horsepower is needed to overcome the wind drag on this car if it is traveling 50 miles per hour? Algebra II

Word problems: The value of Jennifer’s stock portfolio is given by the function v(t) = 50 + 73t − 3t2 , where v is the value of the portfolio in hundreds of dollars and t is the time in months. How much money did Jennifer start with? When will the value of Jennifer’s portfolio be at a maximum? Algebra II

Closure: Exit Pass The path of a basketball thrown at an angle of 45 can be modeled by y = -0.02x2 + x + 6. Denecia catches the ball at its highest point to complete an alley-oop. At what height does she catch the ball and and how long will it be before she catches the ball? OG Volleyball team charges $3 per ticket. At that rate they sell 100 tickets. For every $.50 decrease in price they would sell 20 more tickets. What price should they charge to maximize revenue? What is the maximum revenue? The Warrior Beat has an ad that is 12 by 2 inches. They want to put a gold border of uniform width around the ad to make it stand out. They have 75 square inches to fill for the ad and border. How wide should the border be? Algebra II

Closure: Graph each. 1. y = -½(x – 3)(x + 1) 2. y = 2(x + 3)2 – 2 3. y = –x2 + 4x – 2 Algebra II