Quadratic Functions (Section 2-1)

Transcription

1 Quadratic Functions (Section 2-1)

2 Section 2.1, Definition of Polynomial Function

3 f(x) = a is the constant function f(x) = mx + b where m 0 is a linear function f(x) = ax 2 + bx + c with a 0 is a quadratic function The graph of a quadratic function is a U-shaped curve called a parabola. The axis of symmetry of the parabola is the line about which the parabola is symmetric. The vertex of the parabola is the point where the axis of symmetry intersects the parabola. oif a > 1, the graph opens upwards. If a < 1, the graph opens downwards. o a > 1 is a vertical stretch and 0 < a < 1 is a vertical shrink.

4 Example 1 Describe how the graph of each function is related to the graph of y = x 2? a) 1 f ( x) x 3 2 b) g( x) 2x 2 c) h( x) x 2 1 d) k( x) x

5 Standard Form of the Quadratic Equation f(x) = a(x-h) 2 + k a 0, where the Axis of Symmetry: line x = h and the Vertex: (h, k) To find the x-intercepts of the graph solve the equation ax 2 + bx + c = 0.

6 Sketch the graph of the quadratic function. Identify the vertex and x-intercepts. Example 2 f(x) = 64 x 2

7 Sketch the graph of the quadratic function. Identify the vertex and x-intercepts. Example 3 f(x) = ( x 3) 2 + 6

8 Sketch the graph of the quadratic function. Identify the vertex and x-intercepts. Example 4 f(x) = x 2 6x + 8

9 Sketch the graph of the quadratic function. Identify the vertex and x-intercepts. Example 5 f(x) = x 2 +6x 8

10 Sketch the graph of the quadratic function. Identify the vertex and x-intercepts. Example 6 f(x) = 2x 2 + 8x + 7

11 HW # pg 99 (1-13 odd, odd)

12 Write the standard form of the quadratic function that has the indicated vertex and whose graph passes through the given point. Verify your result with a graphing utility. Example 7 Vertex: (1, 2) Point: (3, -6)

13 Write the standard form of the quadratic function that has the indicated vertex and whose graph passes through the given point. Verify your result with a graphing utility. Example 8 Vertex: (-4, 11) Point: (-6, 15)

14 Determine the x-intercept(s) of the graph visually. How do the x-intercepts correspond to the solutions of the quadratic equation when y = 0? Example 9 y = x 2 + x 2

15 Find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given x-intercepts. (There are many correct answers.) Example 10 5,0, 2 2,0

16 Find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given x-intercepts. (There are many correct answers.) Example ,0, 3,0

17 Minimum and Maximum values of Quadratic Function ax 2 + bx + c = 0 1) If a > 0, f has a minimum value at b x 2a 2) If a < 0, f has a maximum value at b x 2a

18 Example 12 (example 5 in book) A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet per second and at an angle of 45 with respect to the ground. The path of the baseball is given by the function f(x) = x 2 + x + 3, where f(x) is the height of the baseball (in feet) and x is the horizontal distance from home plate (in feet). What is the maximum height reached by the baseball?

19 Example 13 A football is thrown at a point 6 feet above the ground at a velocity of 60 feet per second and at an angle of 45 with respect to the ground. The path of the baseball is given by the function f(x) = x 2 + x + 6, where f(x) is the height of the football (in feet) and x is the horizontal distance from the quarterback(in feet). What is the maximum height reached by the football?

20 Example 14 (example 6 in book) A soft drink manufacturer has daily production costs of C(x) = 70, x x 2 Where C is the total cost (in dollars) and x is the number of units produced. Estimate numerically the number of units that should be produced each day to yield a minimum cost.

21 Example 15 A local newspaper has daily production costs of C = x x 2 where C is the total cost (in dollars) and x is the number of newspapers printed. How many newspapers should be printed each day to yield a minimum cost?

22 Find the value of b such that the function has the given maximum or minimum value. Example 16 f(x) = -x 2 + bx 16 Maximum value: 48

23 HW # pg (29 47 odd, 55, 57, 59, 65, 67, 69, 73, 75)