G RAPHING A Q UADRATIC F UNCTION A quadratic function has the form y = ax 2 + bx + c where a  0. The graph is “U-shaped” and is called a parabola. The highest or lowest point on the parabola is called the ver tex. In general, the axis of symmetry for the parabola is the vertical line through the vertex.

G RAPHING A Q UADRATIC F UNCTION The origin is the lowest point on the graph of y = x 2, and the highest point on the graph of y =  x 2. These are the graphs of y = x 2 and y =  x 2. The origin is the vertex for both graphs. The y -axis is the axis of symmetry for both graphs.

G RAPHING A Q UADRATIC F UNCTION THE GRAPH OF A QUADRATIC FUNCTION The axis of symmetry is the vertical line x = –. b 2a2a The graph of y = a x 2 + b x + c is a parabola with these characteristics: The parabola opens up if a > 0 and opens down if a < 0. The parabola is wider than the graph of y = x 2 if a 1. The x-coordinate of the vertex is –. b 2a2a C ONCEPT S UMMARY

Graphing a Quadratic Function Graph y = 2 x 2 – 8 x + 6 S OLUTION Note that the coefficients for this function are a = 2, b = – 8, and c = 6. Since a > 0, the parabola opens up. x = – = – = 2 b 2 a2 a – 8 2(2) The x-coordinate is: Find and plot the vertex. y = 2(2) 2 – 8 (2) + 6 = – 2 The y-coordinate is: So, the vertex is (2, – 2). (2, – 2)

Graphing a Quadratic Function Graph y = 2 x 2 – 8 x + 6 S OLUTION (2, – 2) Draw the axis of symmetry x = 2. Plot two points on one side of the axis of symmetry, such as (1, 0) and (0, 6). (0, 6) (1, 0) Use symmetry to plot two more points, such as (3, 0) and (4, 6). (4, 6) (3, 0) Draw a parabola through the plotted points.

G RAPHING A Q UADRATIC F UNCTION VERTEX AND INTERCEPT FORMS OF A QUADRATIC FUNCTION FORM OF QUADRATIC FUNCTIONCHARACTERISTICS OF GRAPH Vertex form: Intercept form: y = a (x – h) 2 + k y = a (x – p )(x – q ) For both forms, the graph opens up if a > 0 and opens down if a < 0. The vertex is (h, k ). The axis of symmetry is x = h. The x -intercepts are p and q. The axis of symmetry is half- way between ( p, 0 ) and (q, 0 ). C ONCEPT S UMMARY

Graphing a Quadratic Function Graph y = – (x + 3) S OLUTION The function is in vertex form y = a (x – h) 2 + k. a = –, h = – 3, and k = a < 0, the parabola opens down. first plot the vertex (h, k) = (– 3, 4). ( – 3, 4)

Graphing a Quadratic Function Graph y = – (x + 3) S OLUTION Draw the axis of symmetry x = – 3. Plot two points on one side of it, such as (–1, 2) and (1, – 4). Use symmetry to complete the graph. (– 3, 4) (1, – 4) (–1, 2) (– 7, – 4) (– 5, 2)

Graphing a Quadratic Function Graph y = – ( x +2)(x – 4) S OLUTION The quadratic function is in intercept form y = a (x – p)(x – q), where a = –1, p = – 2, and q = 4. The x-intercepts occur at (– 2, 0) and (4, 0). (– 2, 0) (4, 0) (– 2, 0) (4, 0)The axis of symmetry lies half-way between these points, at x = 1.

Graphing a Quadratic Function Graph y = – ( x +2)(x – 4) S OLUTION (– 2, 0) (4, 0) So, the x -coordinate of the vertex is x = 1 and the y -coordinate of the vertex is: y = – (1 + 2) (1 – 4) = 9 (1, 9)

Graphing a Quadratic Function