WARM UP 1.Use the graph of to sketch the graph of 2.Use the graph of to sketch the graph of

GRAPHS OF QUADRATIC FUNCTIONS

OBJECTIVES Sketch a graph showing vertical stretching or shrinking. Sketch a graph showing horizontal stretching & shrinking. Graph and determine its characteristics. Solve problems using quadratic functions.

GRAPHS Compare the graphs of y = f(x), y = 2f(x), and y = 1/2f(x). The graph of y = 2f(x) looks like that of y = f(x), but is stretch vertically. The graph of y = 1/2f(x) is flattened, or shrunk vertically.

VERTRICAL STRETCHING Consider any equation y = f(x). If we multiply f(x) by 2, then every function value is doubled. This has the effect of stretching the graph away from the horizontal axis. This is true for any constant between 0 and 1. Multiplying f(x) by ½ will halve every function value, thus shrinking the graph towards the horizontal axis. This is true for any constant between 0 and 1.

GRAPHS Now compare the graphs of y = f(x), y = -2f(x), and y = -1/2 f(x). When we multiply by a negative constant, the graph is reflected across the x-axis, and is also being stretched or shrunk. Note that multiplying f(x) by -1 has the effect of replacing y by –y, and that we obtain a reflection without stretching or n shrinking.

VERTRICAL STRETCHING Consider y = f(x). Multiply f(x) by a constant c. We then have y = c  f(x). This is equivalent to Thus, in an equation of any relation, dividing by 2 will stretch the graph in the y – direction. Similarly dividing y by ½ will shrink the graph in the y-direction.

THEOREM 9-7 In an equation of a relation, dividing y by a constant c does the following: A.If, the graph is stretched vertically. B.If, the graph is shrunk vertically. C.If c is negative, the graph is also reflected across the x-axis.

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 Graphs of quadratic functions are called parabolas, Line of symmetry

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

TRY THIS… a.Graph f(x) = -1/4x b.What is the line of symmetry? c.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 a.Graph f(x) = -2(x – 1) b.What is the line of symmetry? c.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… a.Graph f(x) =3(x – 2) b.What is the line of symmetry? c.What is the vertex? The line of symmetry is x = 2. The vertex is (2, 0).