When graphing quadratic equations, there are a few key terms that we can identify to help us graph quadratic functions more efficiently. Instead of plugging in values, we can find specific characteristics such as the y-intercept, x-intercepts, direction, shape, vertex and axis of symmetry. When graphing quadratic equations, it is always best to initially graph the function by hand and then check your work using a graphing calculator if able.
When it comes time to graph parabolas you could always make a table of values and plug in the x points one by one to find their y points. It takes a long time though and sometimes there shortcuts or other points you could use to help you do your graph more efficiently. Also if you guys are lucky you might have access to a graphing calculator but only use that to check your work, you don't want to rely on the graphing calculator when you should know how to do this stuff by hand. There's a whole bunch of information that's going to help you to graph your parabolas and I listed a bunch of it here.
The first thing you want to want to look for is the y intercept, remember the y intercept is found by letting x=0. That's usually a really quick one and a great way to find at least one point on your parabola.
The x intercept has to do with all this stuff you've been studying in the study of quadratics we're talking about to find the x intercepts you could use the discriminant to tell you how many x intercepts there are and then to find the actual x intercepts you have a choice of what method to use; factoring, the quadratic formula, taking square roots or completing the square those are all options to you. They all take different amounts of time some other more easier for certain problems than others, so something you're going to want to be practicing is being able to tell which method to use in which circumstances and your teachers hopefully helping you with that in class.
Another thing you're going to want to look at is whether or not the parabola opens upward or opens down. Some people think of this as like smiley face like a positive a value or sad face a negative a value that's how you tell if the parabola opens up or opens down. This often shows up on multiple choice tests a lot of times teachers will give you the correct x and y intercepts and they'll give you like one graph where the parabola opens up one graph where the parabola opens down the way you'll be able to tell is by looking at a your lead coefficient if a is positive the parabola opens up I'll write that here, if a is positive, it opens down if a is negative.
Okay skinny or wide parabola, if your a value is a whole number like 2, 3, 4 something like that or it could be -2, -3, -4 then it's going to be a skinny parabola. If you have a fraction a, it's going to be a wide parabola, fraction a value, remember a means the lead coefficient it's the coefficient in front of the x squared term.
The last thing you might want to look at is the vertex and reminder the way to find the vertex is by first finding the x value with x is equal to b over 2a. Once you find your x number substitute that value back into the function to find the y value. The vertex is the point where your parabola either reaches it's bottom or it's top it's really important point it also helps you to find the axis of symmetry, remember the axis of symmetry is the equation x equals negative b over 2a and it goes, it's the vertical line that goes right through your vertex. That will also help you when you're drawing your graph.
Last but not least if you still feel like you've tried all these things and you still don't have a very good idea what your parabola looks like try making a table of values you can always use this method and as long as you're careful with order of operations those points will help you draw the parabola but these things are the most important elements of the quadratic function, so it's important that you guys know how to do all of these skills as well.