Need to graph parabolas (quadratic equations) on a test but don’t know how?  Here’s a way you can create a detailed graph by hand while giving all the hard work to the TI-84.  How to graph quadratic equations has never been easier!

Let’s say we want to graph the function f(x) = (-1/3)x2 + 2x – 3.

1.)   Press the Y= button and enter in the right side of the equation. Don’t forget to square the first x!

2.)  Press GRAPH to view your parabola.

3.)   In order to graph parabolas well, we need to find the vertex. I cover this in more detail in my video and TI-84 How To article entitled Graphing Calculator: How To Find the Vertex of a Parabola.
a.)  Press 2nd, Calc.
b.)  Select either maximum or minimum.  In this case, we will select maximum, because the vertex occurs on the highest point on the parabola.
c.)   Choose your left down, right down, and then “guess” where your vertex is by moving the cursor as close as you can to it. Then press enter.

The vertex of this function is (3, 0).

Now we need to find the coordinates of some other points on the parabola, so that we can draw an accurate looking graph.

We could do this by hand by substituting values of x into the equation and finding their respective y-values.

However, we can save a lot of work by calculating a table of values on the TI-84.  How to do this is in my explanation below.

4.)   Simply press 2nd, GRAPH,  and the calculator will show you the table of values for this function.

5.)   Now we want to select some points from this table to graph. I recommend selecting  3 points to the left of the vertex, and 3 points to the right of the vertex. In this example, I shall pick (2, -1/3), (1, -4/3), (4, -1/3), (5, -4/3), (6, -3), (0, -3).

6.)  Now, all we need to do is graph these points on graph paper. Note that some of these points have fractional Y values. To plot these points, you need to estimate where the point with a  y-value of one third and four thirds would be.

7.)   Once you have plotted the points, connect the dots with the curve.

Now you have your graph of a parabola, which you did with the help of your TI-84!  How to solve more math problems can be found in my TI-84 How To blog.

Here’s another trick you can do on your graphing calculator: how to find the vertex of a parabola.

Let’s say we want to find the vertex of f(x) = x2 – 3x + 6

1.)  First, you need to graph the equation.  Press Y= and enter in the right side of this equation.  Then hit GRAPH.

2.)  Note that the vertex at the bottom of this parabola.

3.)  Press 2nd, then CALC.  Since the vertex occurs at a minimum value, we’ll select option 3, “minimum.”

4.)  The graphing calculator will now ask us to define our “left bound.”  To do this, just make sure the cursor is somewhere to the immediate left of the vertex, and press ENTER.

5.)  Now, the graphing calculator will ask for a right bound.  Move the cursor somewhere to the immediate right of the vertex, and press ENTER.   You should see two triangles now– one at the left bound, and one at the right bound.

6.)  Now the graphing calculator will ask you to “guess” where the vertex is.  Move your cursor as close as you can to the vertex, and press enter.

Sometimes the calculator will give a value that’s just a little off.  In this case, the x-value of the vertex is not 1.5000018, but 1.5, or 3/2.  The answer is (1.5, 3.75), or (3/2, 15/4).

Note that sometimes the vertex will occur at a maximum, for example, f(x) = -0.5×2 + x – 2.

In this case, when we graph the parabola, the vertex will be at the top of the parabola.  When we go to 2nd, CALC, we will choose option 4, “maximum.”  After that, we’ll go through the same set of steps of choosing a left bound, right bound, and guessing.

The answer is (1, -3/2)

On your own, find the vertex of f(x) = -2×2 – 2x – 2.
The answer is at the end of the video above.

To understand this even better, watch the video about the TI-84: how to find the vertex.