WARM UP Use the graph of to sketch the graph of

WARM UP Tell whether the graphs of the following are symmetric with respect to the x-axis, y-axis, the origin or none. y = x x + y = 2 x + 2y = 3 Symmetric with respect to the y-axis. Symmetric with respect to the x-, y-axis & origin. None

GRAPHS OF QUADRATIC FUNCTIONS

OBJECTIVES Graph and determine its characteristics. Solve problems using quadratic functions.

QUADRATIC FUNCTIONS Graph the equation , and on the same set of axes. Study the graph that you have drawn. In the graphs of the equation of the form , what effect does changing the value of a have on the graph? Now graph the equation , , and . Use a new set of axes. Again, study the graphs that you have drawn. In graphs of equations of the form , what effect does h have on the graph?

DEFINITION A quadratic function is a function that can be described as: , where a ≠ 0 Line of symmetry Graphs of quadratic functions are called parabolas,

VERTEX Consider the graph of f(x) = x . The function is even because f(x) = f(-x) for all x. Thus the y-axis is the line symmetry. The point (0, 0), where the graph crosses the line of symmetry is called the vertex of the parabola. Next we consider f(x) = ax . By Theorem 9-7, we know the following about its graph. Compared with the graph of f(x) = x 1. If , the graph is stretched vertically. 2. If , the graph is shrunk vertically. 3. If a < 0, the graph is reflected across the x-axis.

EXAMPLE 1 Graph f(x) = 3x What is the line of symmetry? What is the vertex? The line of symmetry is the y-axis. The vertex is (0, 0)

TRY THIS… Graph f(x) = -1/4x What is the line of symmetry? What is the vertex? The line of symmetry is the y-axis. The vertex is (0, 0)

MORE GRAPHS In f(x) = , let us replace x by x – h. By Theorem 9-6, if h is positive, the graph will be be transferred to the right. If h is negative, the translation will be to the left. The line, or axis, of symmetry, and the vertex will also be translated the same way. Thus for f(x) = a(x – h) , the axis of symmetry is x – h and the vertex is (h, 0). Compare the graphs of f(x)=2(x +3) to the graph of f(x) =2x Vertex (-3, 0) Line of symmetry x = -3

EXAMPLE 2 Graph f(x) = -2(x – 1) What is the line of symmetry? What is the vertex? We obtain the line of symmetry from the equation x – 1 = 0; the line of symmetry is x = 1. The vertex is (1, 0) Vertex (1, 0) Line of symmetry x = 1

TRY THIS… Graph f(x) =3(x – 2) What is the line of symmetry? What is the vertex? The line of symmetry is x = 2. The vertex is (2, 0).

CH. 9.4 HOMEWORK Textbook pg. 402 #2, 6, 10, 16, & 20