Section 9.1 Identifying Quadratic Functions Section 9.2 Characteristics of Quadratics

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1 1 Algebra 1, Quadratic Notes Name Learning Targets: Section 9.1 Identifying Quadratic Functions Section 9.2 Characteristics of Quadratics Identify quadratic functions and determine whether they have a minimum or a maximum. Graph a quadratic function and determine it's domain and range. Find the zeros of a quadratic function from it's graph. Find the axis of symmetry and the vertex of a parabola. Key Terminology: Quadratic Function (pg. 590) Success Criteria Identify quadratic functions. Use a table to graph a quadratic function. Identify the vertex and minimum or maximum. Find the domain and range of a quadratic function. Find zeros of quadratic functions from graphs. Find the axis of symmetry by using zeros and a formula. Find the vertex of a parabola using a formula. Parabola (pg. 591) Vertex (pg. 592) Minimum (pg. 593) Maximum (pg. 593)

2 2 Zero of a Function (pg. 599) Axis of Symmetry (pg. 600) Examples: 1. Tell whether the function is quadratic. Explain. a. b. y = 7x + 3 c. y 10x 2 =9 d. {(-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4)}

3 3 2. Graph the following quadratic equations. a. y= 1 3 x2 b. f % x&= 3x 2 $1 3. Identify if the vertex is a maximum or minimum value. Explain. a. y 1 3 x2 =x 3 b. f % x&= 4x 2 x$1 c. y=5x 3x 2

4 4 4. Identify the vertex of each parabola. Then give the minimum or maximum value of the function. 5. Find the domain and range. a. b. 6. Find the zeros of the quadratic function from its graph. Check your answer. a. y=x 2 2x 3 b. y=x 2 $8x$16

5 5 6. (cont.) Find the zeros of the quadratic function from its graph. Check your answer. c. y= 2x 2 2 d. y=x 2 6x$9 7. Find the axis of symmetry of each parabola. a. b.

6 6 c. d. 8. Find the axis of symmetry of the graph of : a. y= 3x 2 $10x$9 b. f % x&=2x 2 $ x$3

7 7 9. Find the vertex. a. y=0.25x 2 $2x$3 b. y= 3x 2 $6x The graph of f(x) = 0.06x x can be used to model the height in meters of an arch support for a bridge, where the x-axis represents the water level and x represents the distance in meters from where the arch support enters the water. Can a sailboat that is 14 meters tall pass under the bridge? Explain. 11. The height of a small rise in a roller coaster track is modeled by f(x) = 0.07x x , where x is the distance in feet from a supported pole at ground level. Find the height of the rise.

8 8 Section 9.3 Notes: Graphing Quadratic Functions Learning Targets: Graph a quadratic function in the form of y=ax 2 $bx$c. Success Criteria Use the axis of symmetry, vertex and the y-intercept to graph a quadratic function. Use characteristics of quadratic functions to find answers to applications. Examples: 1. Graph y=3x 2 6x$1 without a calculator. Axis of Symmetry: Vertex: Y-intercept: Two other points: and 2. Graph y$6x= x 2 $9 without a calculator. Axis of Symmetry: Vertex: Y-intercept: Two other points: and

9 9 3. Graph y=2x 2 $6x$2 without a calculator. Axis of Symmetry: Vertex: Y-intercept: Two other points: and 4. The height in feet of a basketball that is thrown can be modeled by f % x&= 16x 2 $32x, where x is the time in seconds after it is thrown. Find the basketball s maximum height and the time it takes the basketball to reach this height. Then find how long the basketball is in the air.

10 5. As Molly dives into her pool, her height in feet above the water can be modeled by the function f %x&= 16x 2 $24x, where x is the time in seconds after she begins diving. Find the maximum height of her dive and the time it takes Molly to reach this height. Then find how long it takes her to reach the pool. 10

11 11 Learning Targets: Section 9.4 Notes: Success Criteria Graph and transform quadratic functions. Compare widths of parabolas using the standard form equation. Compare graphs of quadratic functions when given an equation in standard form. Key Terminology: None The quadratic parent function is f(x) = x 2. The graph of all other quadratic functions are transformations of the graph of f(x) = x 2. Examples: The value of a in a quadratic function determines not only the direction a parabola opens, but also the width of the parabola. 1. Order the functions from narrowest graph to widest. a. f(x) = 3x 2, g(x) = 0.5x 2 b. f % x&=x 2, g % x&= 1 2 x 2, h% x&= 2x 2

12 12 c. f %x&= x 2, g% x&= 2 3 x2 d. f % x&= 4x 2, g %x&=6x 2, h%x&=0.2x 2 2. Compare the graph of the function with the graph of f(x) = x 2. a. g% x&= 1 4 x2 $3 b. g% x&=3x 2

13 13 c. g % x&= x 2 4 d. g% x&= 1 2 x2 $2 The quadratic function h(t) = 16 t 2 + c can be used to approximate the height h in feet above the ground of a falling object t seconds after it is dropped from a height of c feet. This model is used only to approximate the height of falling objects because it does not account for air resistance, wind, and other real-world factors. 3. Two identical softballs are dropped. The first is dropped from a height of 400 feet and the second is dropped from a height of 324 feet. a. Write the two height functions and compare their graphs. b. Use the graphs to tell when each softball reaches the ground. 4. Two tennis balls are dropped, one from a height of 16 feet and the other from a height of 100 feet. a. Write the two height functions and compare their graphs. b. Use the graphs to tell when each tennis ball reaches the ground.