Unit
Quadratic Equations and Functions
MA, Stanford University
Teaching in the San Francisco Bay Area
Alissa is currently a teacher in the San Francisco Bay Area and Brightstorm users love her clear, concise explanations of tough concepts
We have rational functions whenever we have a fraction that has a polynomial in the numerator and/or in the denominator. An excluded value in the function is any value of the variable that would make the denominator equal to zero. To find the domain, list all the values of the variable that, when substituted, would result in a zero in the denominator.
Any time you're asked to graph a quadratic equation, it's really important that you get a nice smooth shape u or parabola but first make sure you know what a quadratic means. A quadratic equation is something where the highest exponent on x is 2. You have an x squared and that's your largest exponent, so anytime you see an x squared it's not going to be a line it's going to be curve u-shape. Sometimes the u is going to open up, sometimes the u is going to open down so we call it parabola.
Another thing to look for is to look for symmetry in the table and the graph. You guys probably already know what symmetry means some shapes you know that are symmetrical are like a heart or a circle or maybe you know the human face is symmetrical what it means is like one half is exactly the same as the other half, symmetry, so when you're going to graph your parabola symmetry is going to be a trick that will help you. When you're looking at your parabola there's a couple things to keep in mind, there is some formal vocabulary that you'll learn more about later but the first thing is that the axis of the symmetry is the line that cuts the parabola in half it's the same on both sides that's the axis of symmetry.
Another thing that's important is this point that's either the top of a parabola that opens downward or the bottom of this guy that's called the vertex and the vertex is the place where your y values turn around that'll make more sense once you start practicing graphing parabolas, so in Math any time you're asked to graph anything whether it's a [IB] or a line or an absolute value or curvy thing or whatever you can always make a table of values. What that means is you pick some x numbers then you substitute them in to your equation one at a time, it's always a good idea to use some negatives and positives we're going to do that here. I'm going to start with negative 3, your teacher might tell you what she or he wants you to use for these, I'm going to go from negative 3 to 3. I just picked that it's always a good idea to use some negative and positives.
Then what I'm going to do is substitute my x value into here one at time like if x was negative 3, y would be negative 3 times itself which is positive 9, negative 2 times itself is positive 4 1, 0, 1, 4 do you guys notice a pattern here? Looking at my table this is what I can see the symmetry 1 shows up on both sides 4 shows up on both sides so I'm predicting that that's going to be 9 just based on symmetry and you can check 3 times itself is indeed 9, so this symmetry helps me figure out my table a little bit more quickly. Also this thing right here that's in the middle of this symmetry you'll start seeing is the vertex. What I'm going to do is put these points on the graph and show you what the parabola u shape looks like. Again any time you graph a quadratic equation, your answer should be a smooth u shaped which we call a parabola, so I had -3, 9 1, 2, 3, 4, 5, 6, 7, 8, 9. -2, 4 -1, 1 to be honest with you guys I don't like graphing parabolas very much because they take a while.
But here's where the shortcuts come in handy, I've reached my vertex I've reached the place in my table where I started drawing the symmetry so now instead of having to count every single point I'm going to use symmetry. I know that this vertical line right here shoop is the place that my parabola folds around it's symmetrical along that vertical line, so this point I'm going to know it's symmetrical across, this point is symmetrical across, this point is symmetrical across, 3 boxes that way 3 boxes that way and equally high. Those were the points I had in my table I just didn't want to keep looking back and forth back and forth back and forth I just use symmetry to help me graph where those points would be.
The last step is to make sure you're drawing a curvy u shape it's not a v, v is for absolute value, we want a parabola means it has to have a curvy bottom also make sure you put arrows on the end because this parabola goes out now forever and ever and ever. So that's it for this problem but when you guys are graphing quadratics please remember you should always have a parabola, it should always be curvy you don't want straight line and never want something funky like this was the point out there like that would tell me that was an incorrect value within my table, so use symmetry use shortcuts if you want to you don't have to they just help your problems go a little more quickly.