Quadratic functions graph as a parabola. All of the points that make up the parabola have (x,y) coordinates that are solutions to the quadratic function of that parabola. There are an infinite number of points making up the parabola. The parabola is either concave up or concave down. The parabola always has a y-intercept, an axis of symmetry, and a vertex point. A parabola has one of the following:

(a) TWO x-intercepts.

(b) ONE x-intercept.

(c) NO x-intercepts.

 

These components are graphed in red in the following two diagrams and all other points of the parabola are graphed in blue.

Quadratic functions always have one term with the exponent equal to 2 and no terms with an exponent higher than 2. The standard form for a quadratic function is

f(x) = ax² + bx + c

where the coefficients a, b, and c are constants that are Real Numbers and a ≠ 0.

Valid (x, y) coordinates of the points that make up the parabola may be determined in different ways, two of which are:

Select any value for x, substitute this value into the function, and solve the equation for y. This produces the (x, y) coordinate for one point on the parabola. Any Real Number may be selected for x to determine an (x, y) coordinate of a point on the parabola. Both positive and negative values should be selected for x in locating points of the parabola.

Example:

We will use the parabolic function f(x) = x² -6x + 9.

        Select x = 2.

        Substitute 2 for x in the parabolic function and solve for y:

                y = x² - 6x + 9

                y = (2)² - ( 6 • 2 ) + 9

                y = 4 – 12 + 9

                y = 1

        So the point ( 2, 1 ) is a point on the parabola.

        Using x = 3, we find the point ( 3, 0 ) .

        Using x = 0, we find the point ( 0, 9 ) .

        Using x = 6, we find the point ( 6, 9 ) .

        Using x = 4, we find the point ( 4, 1 ) .

Plotting these points on the parabola and connecting the points, we obtain the following parabola:

Example1:

We will use the same parabolic function f(x) = x² -6x + 9.

        Press the "Y=" key on the calculator.

Enter "x² -6x + 9" for "Y1=" . (Note that the quotes are not entered on the calculator.)

Press the "ZOOM" key.

Press the option "9".

(The graph of the parabola is now displayed.)

 

Press the "2^nd" key.

Press the "Table" key.

A table of "X, Y1" values are now displayed. Each  (x, y) value is a point of the parabola.

 

Example2:

To obtain specific (x, y) values another way, after entering the function in "Y1=":

Press the "2^nd" key.

Press the "Quit" key.

Press the "VARS’ key.

Using the right arrow key, cursor across to the "Y-VARS" heading and press the "ENTER" key when the "1:Function" option is highlighted.

Press the "ENTER" key again when the "1:Y1" option is highlighted. Now "Y1" is displayed at the bottom of the screen.

Press the "(" key.

Press the "2" key. (Note that we are indicating that x = 2 and we are looking for the corresponding y-value.)

Press the ")" key.

Press the "ENTER" key.

Now the y-value "1" is now displayed at the bottom of the screen. This is the corresponding value for x = 2. So we have located the point ( 2, 1 ). Continuing with this process, we may enter any value for "x" and locate the corresponding "y" value.

To review the special parts of a parabola, press the option at the top of this screen titled "Parts of a Parabola".