Graphing Quadratic Equations
Graphing quadratic equations requires the culmination of a lot of concepts and procedures. Here, we practice graphing without having to use the quadratic formula, which is very useful for non-factorable quadratic equations. We will learn about that formula in the next section.
Read through the notes, taking notes of your own and trying the examples. Then, watch the videos and try the practice problems.
This is a very important skill, one that will make future math a lot more accessible and easy for you. You should thank yourself for taking the time and effort to dive in here! Be patient, learning always requires dealing with confusion and often frustration to boot!
In this section we will discuss graphing quadratic equations by finding key features, and perhaps using a small t-chart in a couple of special cases. It is difficult to say, this is how you always graph a parabola because of a few things that can happen. However, most of the time, the diagram below will get you a decent graph. When graphing, it is common that you will have two x – intercepts, and a y – intercept and a vertex, that are all at different points.
You simply find the x – intercepts, plot them. Then, you find the y – intercept and plot that point. Then, you find the vertex, and plot that point. Check that the line of symmetry, which passes vertically through the x of the vertex, is exactly half-way between your x – intercepts. Also, make sure that your vertex is either the maximum value (when a is negative), or minimum (when a is positive).
One thing that can help, especially at first, is if you place the letter V next to the vertex. Remember, that is the “turn-around,” point, and each side of the parabola should be symmetrical. The left side and right side should be mirror images of each other.
Here are a few things to keep in mind.
- If b = 0, the vertex and y – intercept are the same location.
- If a > 0, the parabola goes up
- The vertex is the minimum value
- The Range is from the y – coordinate of the vertex to positive infinity
- The vertex is the minimum value
- If a < 0, the parabola goes down
- The vertex is the maximum value
- The Range is from negative infinity to the y – coordinate of the vertex
- If the equation is a perfect square, the x – intercept (only one) will be the same as the vertex
- There can be zero number of x – intercepts. This happens when you find the solutions to f(x) = 0 are Imaginary numbers (square roots of negative numbers)
To find the x – intercepts, solve the equation f(x) = 0, try factoring first if the equation is in standard form. If the equation is in vertex form, use inverse operations. If in standard form and it is unfactorable, try using the quadratic formula (which comes from completing the square).
To find the y – intercept, plug in zero for x, and simplify. Evaluate f(0).
To find the vertex from standard form, use the formula:
Finding the vertex from vertex form is easy, it is just (h, k). Be care of the sign of h.
If you do NOT have x – intercepts, or you would like a few more points to plot, pick values close to the x – coordinate of the vertex and plug them in. For example, if the vertex was (1, 3), and the leading coefficient was positive, there would be zero x – intercepts. You could plug in 2, as in evaluating f(2), and also 3, as in f(3), to find some more points to plot. Those points could then be reflected over the line of symmetry to find additional points to help you make your parabola.
Parabolas: When you sketch the parabola it needs to be smooth and continuous. That is, the near the vertex it is not a V, but more like a U. The sides are curved, not lines, and not parallel, but always expanding.
Using softwartry the graphing of an equation on your, and then plug the equation into something like Desmos.com (or the app on your phone, it’s free), and compare your graph to the computer generated version. This is a great way to check your progress and inform yourself of what’s going well and what mistakes you’re making.