G RAPHING A Q UADRATIC F UNCTION A quadratic function has the form y = ax 2 + bx + c where a  0.

G RAPHING A Q UADRATIC F UNCTION The graph is “U-shaped” and is called a parabola.

G RAPHING A Q UADRATIC F UNCTION The highest or lowest point on the parabola is called the ver tex.

G RAPHING A Q UADRATIC F UNCTION In general, the axis of symmetry for the parabola is the vertical line through the vertex.

G RAPHING A Q UADRATIC F UNCTION These are the graphs of y = x 2 and y =  x 2.

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

G RAPHING A Q UADRATIC F UNCTION The y -axis is the axis of symmetry for both graphs.

THE GRAPH OF A QUADRATIC FUNCTION G RAPHING A Q UADRATIC F UNCTION C ONCEPT S UMMARY 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

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. Graphing a Quadratic Function

Graph y = 2 x 2 – 8 x + 6 x = – = – = 2 b 2 a2 a – 8 2(2) y = 2(2) 2 – 8 (2) + 6 = – 2 So, the vertex is (2, – 2). (2, – 2) The x -coordinate is: The y -coordinate is: Find and plot the vertex.

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

VERTEX AND INTERCEPT FORMS OF A QUADRATIC FUNCTION G RAPHING A Q UADRATIC F UNCTION 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 ).

Graphing a Quadratic Function Graph y = – (x + 3) To graph the function, first plot the vertex (h, k) = (– 3, 4). ( – 3, 4) 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.

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

Graphing a Quadratic Function in Intercept Form Graph y = – ( x +2)(x – 4) The quadratic function is in intercept form y = a (x – p)(x – q), where a = –1, p = – 2, and q = 4. S OLUTION

Graphing a Quadratic Function in Intercept Form Graph y = – ( x +2)(x – 4) The axis of symmetry lies half-way between these points, at x = 1. (– 2, 0) (4, 0) The x-intercepts occur at (– 2, 0) and (4, 0). (– 2, 0) (4, 0)

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

G RAPHING A Q UADRATIC F UNCTION You can change quadratic functions from intercept form or vertex form to standard form by multiplying algebraic expressions. One method for multiplying expressions containing two terms is FOIL. Using this method, you add the products of the First terms, the O uter terms, the Inner terms, and the Last terms.

G RAPHING A Q UADRATIC F UNCTION First F F O O I I L L = x x x 2 5x 3x3x 15( x + 3 )( x + 5 ) = OuterInnerLast FOIL