1 Warm-up Factor the following x 3 – 3x 2 – 28x 3x 2 – x – 4 16x 4 – 9y 2 x 3 + x 2 – 9x - 9

Graphing Quadratic Functions Section 2-1

3 Objectives I can graph quadratic functions in vertex form. I can convert to vertex format by completing the square I can find the equation of a quadratic given the vertex point and another point.

4 Applications Many applications that are used to model consumer behavior. Example: Revenue generated from manufacturing handheld video games

5 Quadratic function Let a, b, and c be real numbers a  0. The function f (x) = ax 2 + bx + c is called a quadratic function. The graph of a quadratic function is a parabola. Every parabola is symmetrical about a line called the axis (of symmetry). The intersection point of the parabola and the axis is called the vertex of the parabola. x y axis f (x) = ax 2 + bx + c vertex

6 Quadratic Key Terms Vertex: The vertex is the point of intersection between the Axis of Symmetry and the parabola. Axis of Symmetry: This is the straight line that cuts the parabola into mirror images. Solutions (Also called Roots): These are the x-coordinates of the points where the parabola intersects the x-axis.

7 Two Real Solutions

8 One Real Solution

9 No Real Solutions

10 y=ax 2 +bx+c One key point to graph is always the y- intercept. This is a point that has the x value equal to zero (0, b) If we let “x” be zero if the equation, then we see that the y-intercept is just the value of “c” in the equation. Example: y = 2x 2 – 3x + 4 The y-intercept is (0, 4)

11 Solutions versus Intercepts Lets look at the differences in these vocabulary terms: Solutions, roots, zeros, x-intercepts If the problem asks you to find solutions or roots your answer should be in the format x = ?? Or {2, 3, 5}. We just list the x-value If the problem asks you to find zeros or x-intercepts, your answer should be in ordered pair format: (2, 0) (-3, 0)

12 Leading Coefficient The leading coefficient of ax 2 + bx + c is a. When the leading coefficient is positive, the parabola opens upward and the vertex is a minimum. When the leading coefficient is negative, the parabola opens downward and the vertex is a maximum. x y f(x) = ax 2 + bx + c a > 0 opens upward vertex minimum x y f(x) = ax 2 + bx + c a < 0 opens downward vertex maximum

13 5 y x Simple Quadratic Functions The simplest quadratic functions are of the form f (x) = ax 2 (a  0) These are most easily graphed by comparing them with the graph of y = x 2. Example: Compare the graphs of, and

14 Standard Vertex Format The format for any parabola with an equation ax 2 + bx + c = 0 can be written in the following standard vertex format: y = a(x – h) 2 + k Where the Vertex is (h,k) and the Axis of Symmetry is x = h If a > o, then the parabola opens upward If a < o, then the parabola opens downward

15 Example 2: y = x 2 – 4x + 5 y = x 2 – 4x + 5 (Now complete the square) y = (x 2 – 4x + ___) _____ -4/2 = -2 (-2) 2 = 4 y = (x 2 – 4x + 4) + 5 – 4 y = (x – 2) Vertex is (2, 1), Axis of Sym: x = 2, a > 0, so parabola turns upward

16 Example 3: y = -5x x y = -5x x (1 st factor –5 from 1 st two terms) y = -5(x 2 – 16x) (Now complete square) -16/2 = -8 (-8)2 = 64, y = -5(x 2 – 16x + 64) (WHY +320??) y = -5(x – 8) Vertex is (8, 1), Axis of Sym: x = 8, a < 0, so parabola turns downward

17 x y 4 4 Vertex and x-Intercepts Example: Graph and find the vertex and x-intercepts of f (x) = –x 2 + 6x + 7. f (x) = – x 2 + 6x + 7 original equation f (x) = – ( x 2 – 6x) + 7 factor out –1 f (x) = – ( x 2 – 6x + 9) complete the square f (x) = – ( x – 3) standard form a < 0  parabola opens downward. h = 3, k = 16  axis x = 3, vertex (3, 16). Find the x-intercepts by solving –x 2 + 6x + 7 = 0. (–x + 7 )( x + 1) = 0 factor x = 7, x = –1 x-intercepts (7, 0), (–1, 0) x = 3 f(x) = –x 2 + 6x + 7 (7, 0)(–1, 0) (3, 16)

Vertex (3, 1) and Point (6, 3) 18

19 Homework WS 3-2