The Graph of the Parabola
Gloria L. Jones
Assignment II (click here for graphing calc. File)
Examine: We will explore the quadratic functions defined by (x)=ax²+bx+c, (a0) and a, b and c are constants. We will see that the graph of any quadratic function is a curve called the parabola which is similar or identical in shape to the graph of y=x². What happens when a=0? When a=0 the function becomes a linear equation that forms a line graph as the graph in green.
First, letıs take a look at the original graph in purple of the quadratic equation y=ax²+bx+c where a, b, and c are equal to 1. Notice that the vertex is not centered on the origin. This is because b and c have a value of 1. Now, letıs look at the graph of the basic equation for a parabola y=ax² in blue where the constants b and c are equal to 0. Notice here that the vertex is centered directly on the origin.
Discussion: From the graphs above in purple, blue, and red, notice that the parabola opens upward. This happens when a>0. From the color key, find that a>0 in each of equations for these graphs. Then look at the graph in light blue where the parabola opens downward. This happens when a<0. For example, the equation for the parabola in light blue is y=(-2)x²+12x-16 where a=(-2) or a<0.
Exploration: We can derive much information from the function of either of the graphs above. Information such the vertex, axis of symmetry, maximum or minimum value of , and x- and y-intercepts can be computed from the equation. Letıs take the function of the graph in light blue and further explore and compute specification of this graph. Recall the function (x)=(-2)x²+12x-16 for the light blue parabola.