Graphing quadratics: standard form

We're asked to graph the following equation y equals 5x squared minus 20x plus 15. So let me get my little scratch pad out. So it's y is equal to 5x squared minus 20x plus 15. Now there's many ways to graph this. You can just take three values for x and figure out what the corresponding values for y are and just graph those three points. And three points actually will determine a parabola. But I want to do something a little bit more interesting. I want to find the places. So if we imagine our axes. This is my x-axis. That's my y-axis. And this is our curve. So the parabola might look something like this. I want to first figure out where does this parabola intersect the x-axis. And as we have already seen, intersecting the x-axis is the same thing as saying when it does this when does y equal 0 for this problem? Or another way of saying it, when does this 5x squared minus 20x plus 15, when does this equal 0? So I want to figure out those points. And then I also want to figure out the point exactly in between, which is the vertex. And if I can graph those three points then I should be all set with graphing this parabola. So as I just said, we're going to try to solve the equation 5x squared minus 20x plus 15 is equal to 0. Now the first thing I like to do whenever I see a coefficient out here on the x squared term that's not a 1, is to see if I can divide everything by that term to try to simplify this a little bit. And maybe this will get us into a factor-able form. And it does look like every term here is divisible by 5. So I will divide by 5. So I'll divide both sides of this equation by 5. And so that will give me-- these cancel out and I'm left with x squared minus 20 over 5 is 4x. Plus 15 over 5 is 3 is equal to 0 over 5 is just 0. And now we can attempt to factor this left-hand side. We say are there two numbers whose product is positive 3? The fact that their product is positive tells you they both must be positive. And whose sum is negative 4, which tells you well they both must be negative. If we're getting a negative sum here. And the one that probably jumps out of your mind-- and you might want to review the videos on factoring quadratics if this is not so fresh-- is a negative 3 and negative 1 seem to work. Negative 3 times negative 1. Negative 3 times negative 1 is 3. Negative 3 plus negative 1 is negative 4. So this will factor out as x minus 3 times x minus 1. And on the right-hand side, we still have that being equal to 0. And now we can think about what x's will make this expression 0, and if they make this expression 0, well they're going to make this expression 0. Which is going to make this expression equal to 0. And so this will be true if either one of these is 0. So x minus 3 is equal to 0. Or x minus 1 is equal to 0. This is true, and you can add 3 to both sides of this. This is true when x is equal to 3. This is true when x is equal to 1. So we were able to figure out these two points right over here. This is x is equal to 1. This is x is equal to 3. So this is the point 1 comma 0. This is the point 3 comma 0. And so the last one I want to figure out, is this point right over here, the vertex. Now the vertex always sits exactly smack dab between the roots, when you do have roots. Sometimes you might not intersect the x-axis. So we already know what its x-coordinate is going to be. It's going to be 2. And now we just have to substitute back in to figure out its y-coordinate. When x equals 2, y is going to be equal to 5 times 2 squared minus 20 times 2 plus 15, which is equal to-- let's see, this is equal to 2 squared is 4. This is 20 minus 40 plus 15. So this is going to be negative 20 plus 15, which is equal to negative 5. So this is the point 2 comma negative 5. And so now we can go back to the exercise and actually plot these three points. 1 comma 0, 2 comma negative 5, 3 comma 0. So let's do that. So first I'll do the vertex at 2 comma negative 5, which is right there. And now we also know one of the times it intersects the x-axis is at 1 comma 0. And the other time is at 3 comma 0. And now we can check our answer. And we got it right.