1 Unit 2 Day 5 Characteristics of Quadratic Functions

2 Warm Up 1.) Jason and Jim jumped off a cliff into the ocean in Acapulco while vacationing. Jason’s height as a function of time could be modeled by the function h(t) = -16t 2 +16t + 480, while Jim’s height could be modeled by h(t) = -16t 2 +12t where t is the time in seconds and h is the height in feet. Whose jump was higher and by how much? 2.) Using the following quadratic, find zeros, y-intercept, vertex, and one other point and the axis of symmetry, then sketch the graph. y = -2x 2 + 4x + 70 Jason jumped 1.75 ft higher with a maximum at (1/2, 484). Jim’s maximum was (3/8, ). Zeros: (-5,0) (7,0) Vertex: (1, 72) y-intercept: (0, 70) Maximum A.o.S. x = 1

3 Homework Answers x(15x +1) x = 0, -1/15 (2x +1)(x + 1) x = -1/2, -1 (2x+3)(2x–3) x = -3/2, 3/2 (0, 0) (-1/15,0) (0, 0) (0, 1) (0,-9) (-1/30, -1/60) (-3/4, -1/8) (0,-9) (-1/2, 0) (-1, 0) (-3/2,0) (3/2, 0) Minimum x=-1/30 x=-3/4 x=0

4 Write the equation for a quadratic function that has the following properties: 4. X intercepts at (4.5,0) and (1,0) and y-intercept at (0,9) 5. X intercepts at (7,0) and (1,0) opening upward 6. X intercepts at (0,0) and (6,0) with a maximum at (3,15) y = 2x 2 – 11x + 9 y = x 2 – 8x + 7 y = -5/3 x x

5 Application Notes p.15 x = 37.5, z = 75 Largest Area = ft 2 A rancher is constructing a cattle pen by the river. She has a total of 150 ft. of fence and plans to build the pen in the shape of a rectangle. Since the river is very deep, she need only fence 3 sides of the pen. Find the dimensions of the pen so that it encloses the maximum area. Area = xzPerimeter: 2x + z = 150 2x + z = 150  z = 150 – 2x (plug into the area) x(150 – 2x) multiplies to 150x – 2x 2 (a quadratic… with a max!) Find the max of y = 150x – 2x 2  (37.5, ) xx z

6 7.) A town is planning a child care facility. The town wants to fence in a playground area using one of the walls of the building. What is the largest playground area that can be fenced in using 100 feet of fencing? Area = xz Perimeter: x + z + x = 100 2x + z = 100  z = 100 – 2x (plug into the area) x(100 – 2x) multiplies to 100x – 2x 2 (a quadratic… with a max!) Find the max of y = 100x – 2x 2 (25, 1250) x = 25, z = 50 Largest Area = 1250 ft 2 xx z

7 Angry Birds Round 1 Show Investigation Notes pages on document camera when reviewing with class.

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9 Angry Birds Round 2 Show Investigation Notes pages on document camera when reviewing with class.

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11 Angry Birds Round 3 For #2, the equation is y = x x Show Investigation Notes pages on document camera when reviewing with class.

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13 Some observations Axis of symmetry: a vertical line that divides the parabola into two congruent halves. It always passes through the vertex of the parabola. The x-coordinate of the vertex is the equation of the axis of symmetry

14 Summary! How to create an equation of a parabola from a graph: 1.Find the zeros. 2.Write the zeros as two binomials and multiply them. 3.Substitute a point to find the a.

15 Quadratic Regression Plug the x values into L1 and the y values into L2. Stat  Calc – QuadReg Enter until you get the equation!

16 Example:

17 Practice

18 Let’s take a quick look at the homework Bridges: Brooklyn Tappan Zee Verrazano

19 Start HW