There are several ways to write the rule of a quadratic function. Each form highlights certain characteristics of the parabola. Standard form is expressed as follows.
When a quadratic function is written in standard form, it's possible to use and to determine characteristics of its graph.
The direction of the graph is determined by the sign of To understand why, consider the quadratic function Since all squares are positive, will always be positive. When is positive, then is also positive. Thus, when moving away from the origin in either direction, the graph extends upward. Similarly, when is negative, will be negative. Thus, the graph will extend downward for all -values.
The -intercept of a quadratic function is given by specifically at This is because substituting into standard form yields the following.
For all quadratic functions, the axis of symmetry will always intersect the parabola at its vertex. Additionally, two points with the same -coordinate will always be equidistant from the axis of symmetry. Move the three points to see how a parabola that passes through them looks.
The function describes the height of the mouth of a tunnel. Here, is the distance from the lower left corner, and both and are in meters. Complete the table of values to graph the function and determine the width and height of the tunnel.
To graph the function, we can plot the points and connect them with a smooth curve.
The graph gives an approximation of the height and width of the tunnel. We can think of the -axis as the ground. Thus, the distance between the -axis and the vertex, which is a maximum, gives the height of the tunnel, and the distance between the zeros gives the width of the tunnel.
Since the axis of symmetry divides the graph into two mirror images, there exists another point on the other side of the axis of symmetry with the same -value as the -intercept. These points are equidistant from the axis of symmetry.
Now, the general shape of the parabola can be seen. Connect the points with a smooth curve.