When graphing and describing the characteristics of a parabola, it is important to know several key pieces of information. The parabola intercepts describe where the parabola intersects the x-axis and the y-axis while the vertex of a parabola is the highest (or lowest) point of the parabola. Knowing the domain and range of a parabola is also helpful when graphing.
Finding the intercepts domain and range and another way to find the vertex a parabola.
So whenever we're finding X intercepts, what we want to do is let Y is equal to 0. So we let Y equal 0. What we end up with is 0 is equal to X squared minus 5X plus 6. So what we end up with is a quadratic equation, which we have the tools to solve.
We can either factor, complete the square, quadratic formula, a number of different ways to solve this out. This particular example factors quite easily. We end up with X minus 2. X minus 3. So we know that our intercepts on R2 and 3. That's easy enough for our X intercepts.
For our Y intercepts, what we want to do is let X equal 0. When we let X equals 0 our first term disappears, our second term disappears, and we're just left with our constant term which in this case is just going to be 6.
Finding the vertex. So two ways of finding the vertex. We can either complete the square, which tends to be pretty involved, or there is a little shortcut which is negative B over 2A is equal to the X coordinate. of the vertex.
So all we have to do is go to our equation. Remember the coefficient on X squared is A. Coefficient on X is B and the constant term is C. So negative B over 2A in this case is just going to be negative, negative 5, 5 over 2 times A which is just 2 times 1. So we find out that our X coordinate is 5 over 2. Okay.
In order to find the other part of our vertex, we just found the X coordinate. We still have to find the Y. All we have to do is plug in the X coordinate into this equation.
This particular example isn't the nicest, because we're dealing with a fraction but it still doesn't matter all we have to do is plug in two and a half. We end up with two and a half squared minus 5 times 2.5 plus 6 and we end up with negative 1.25. So by plugging in that negative B over 2A into the equation we end up with our Y coordinate of our vertici.
To find the domain, domain is value of X that we have to put in, there's no restrictions. We're not dividing by 0. There's no square roots. So X can actually be whatever it wants. So we can call it all reals, negative infinity to positive infinity, different ways of saying the same thing.
The last thing we're looking at is the range of the Y values. This is going to take a little bit of piecing together. So we know we have a parabola. Coefficient on the X squared term is 1, which means our parabola is going to be facing upwards.
So we know we have an upward facing parabola. And we know that our vertex is at a point 5/2s and 1/4. So the Y coordinate of our vertex is going to be the lowest point upward facing parabola. Y coordinate at the bottom. So what we end up is our range being from negative 1/4 to infinity.
Our range actually hits that point. That point is actually a point on the curve. So we then can include negative 1/4 and we're going up. If this was a negative coefficient on the X squared term, I know that the parabola would be going down.
So I would know that I'm going from negative infinity all the way up to that Y coordinate of the vertex. So X intercept and Y intercept are similar to finding any other X intercept, Y intercept, let the other coordinate equal 0 and solve.
Little trick for finding the vertex. Negative V 2 over A plug it in to find the Y coordinate and domain and range and domain is going to be all reallies for parabola, range just consider the vertex and if the graph is going up or down.