Step 2: Determine the x-intercepts if any. The steps for graphing a parabola are outlined in the following example. However, in this section we will find five points so that we can get a better approximation of the general shape. Generally three points determine a parabola. L i n e o f s y m m e t r y V e r t e x x = − b 2 a ( − b 2 a, f ( − b 2 a ) ) We can use the line of symmetry to find the the vertex. Therefore, the line of symmetry is the vertical line x = − b 2 a. To do this, we find the x-value midway between the x-intercepts by taking an average as follows: Using the fact that a parabola is symmetric, we can determine the vertical line of symmetry using the x-intercepts. X - i n t e r c e p t s ( − b − b 2 − 4 a c 2 a, 0 ) and ( − b + b 2 − 4 a c 2 a, 0 ) Therefore, the x-intercepts have this general form: Doing this, we have a 2 + b x + c = 0, which has general solutions given by the quadratic formula, x = − b ± b 2 − 4 a c 2 a. Next, recall that the x-intercepts, if they exist, can be found by setting f ( x ) = 0. In general, f ( 0 ) = a ( 0 ) 2 + b ( 0 ) + c = c, and we have
Given a quadratic function f ( x ) = a x 2 + b x + c, find the y-intercept by evaluating the function where x = 0. Many of these techniques will be used extensively as we progress in our study of algebra. Guessing at the x-values of these special points is not practical therefore, we will develop techniques that will facilitate finding them. In addition, if the x-intercepts exist, then we will want to determine those as well. (also called the axis of symmetry A term used when referencing the line of symmetry.) is the vertical line through the vertex, about which the parabola is symmetric.įor any parabola, we will find the vertex and y-intercept. Lastly, the line of symmetry The vertical line through the vertex, x = − b 2 a, about which the parabola is symmetric. is the point that defines the minimum or maximum of the graph.
The vertex The point that defines the minimum or maximum of a parabola. The x-intercepts are the points where the graph intersects the x-axis. The y-intercept is the point where the graph intersects the y-axis. When graphing parabolas, we want to include certain special points in the graph. Furthermore, the domain of this function consists of the set of all real numbers ( − ∞, ∞ ) and the range consists of the set of nonnegative numbers [ 0, ∞ ). Note that the graph is indeed a function as it passes the vertical line test. and is shared by the graphs of all quadratic functions. This general curved shape is called a parabola The U-shaped graph of any quadratic function defined by f ( x ) = a x 2 + b x + c, where a, b, and c are real numbers and a ≠ 0.