4.1 & 4.2 Graphing Quadratic Functions

Definitions Quadratic Equation: has the form of y=ax2+bx+c and its graph is U-shaped and called a parabola. The lowest or highest point on the graph of a quadratic equation is the vertex. The axis of symmetry for the graph of a quadratic equation is the vertical line through the vertex.

A quadratic functions has the form: We call it’s shape a parabola. The vertex is the “lowest” (or “highest”) point on the parabola. The axis of symmetry is a vertical line through the vertex.

The graph of is a parabola with these attributes: The parabola opens “upward” if a > 0. The parabola opens “downward” if a < 0. The parabola gets “wider” as a gets smaller. RULES TO MEMORIZE!! The parabola gets “narrower” as a get bigger. The x-coordinate of the vertex is The axis of symmetry is the vertical line

Find and plot the vertex: Graph: Solution: Note that the coefficients for this function are a = 2, b = -8, c = 6. Since a > 0, the parabola opens up. Find and plot the vertex: (-) The vertex has coordinates (2,-2) Draw the axis of symmetry x=2 Plot a couple of points on one side of the axes of symmetry, such as (1,0) and (0,6). Use symmetry to plot two more points, such as (3,0) and (4,6)…….

The quadratic function is written in standard form. Two other useful forms for quadratic functions are given below: Vertex and Intercept forms of a Quadratic Functions Form of Quadratic Function Characteristics of Graph The vertex is (h,k) Vertex Form: The axis of symmetry is x = h. Intercept Form: The x-intercepts are p and q. The axis of symmetry is half way between (p,0) and (q,0) For both forms, the parabola opens up if a > 0 and down if a < 0.

Graph this equation given in Vertex Form: Solution: The vertex is (-3,4). The parabola opens down since a < 0. The axis of symmetry is the vertical line x = -3. Plot two points, say (-1,2) and (-5,2).

Graph this quadratic function given in Intercept Form. y = -(x + 2)(x - 4) Solution: The x-intercepts are –2 and 4. The axis of symmetry is x = 1. (WHY?) The vertex has coordinates (1, 9). (WHY?)

You can change quadratic functions from intercept form to standard form by multiplying the algebraic expression. (x + 3)(x + 5) = + 5x +3x +15

Does the parabola open up or down? Is the last equation in standard form?