Today we're to talk about quadratic equations. A quadratic it's just a second degree polynomial with an x squared term at its highest degree term. So we're going to start by looking at what makes an equation quadratic, then we're going to look at some standard forms of quadratic equations, and finally we'll look at the graph of a quadratic equation.
So let's start by looking at an example of a quadratic equation. y is equal to 2x squared plus 3x minus 8. Again, I know this is quadratic because the highest degree is 2. So I'm going to use a table to evaluate different values of x with my equation to find values of y.
And we're going to look at the differences between these values for y. We call this the first differences. And we're going to use those differences to calculate the second differences because for a quadratic, another key definition of a quadratic function is that it has constant second differences. So I'll show you what that means.
So to start we're going to substitute this value for negative 2 in for both x's in my equation. Negative 2 squared is going to give me positive 4 times 2 give me 8. Negative 2 times 3 will give me negative 6. And then bring down my minus 8-- 8 minus 6 minus 8 will give me negative 6.
Do the same thing with the x value of negative 1. Negative 1 squared is positive 1 times 2 is 2. 3 times negative 1 is negative 3 minus 8. This will give me negative 9. Substituting 0 into these two terms, I have 0 for both. So this will give me 0 minus 0 minus 8, which equals negative 8.
Substituting 1 in for both x's, this will give me positive 2. 3 times 1 will give me a positive 3 minus 8 will give me a negative 3. And finally substituting 2 in for my x, 2 squared is 4 times 2 is 8 plus 6 minus 8. It's going to give me a value of 6.
So now I can calculate something called their first differences, which is just going to be the difference between each consecutive pair of y values. So between negative 6 and negative 9, that's going to give me a difference of negative 3, between negative 9 and negative 8, that gives me a difference of 1.
Between negative 8 and negative 3, that gives me a difference of 5. And between negative 3 and 6, that gives me a difference of 9. So I can see that my first differences here are not common. We don't have a common first difference, or common first differences.
If this were a linear function, we would have a common first differences. But since this is quadratic, we're going to look at the second differences, and see that those are common. It's the same number.
So these are our first differences. So if we want to look at our second differences, again, this will be common because this is a quadratic equation. So the difference between negative 3 and 1 is going to give me 4. Between 1 and 5 is also 4, and between 5 and 9, again, we have 4.
So we see that our second differences for this quadratic equation is common. Let's look at three different forms, or ways, of writing a quadratic equation. The first is called standard form. y equals ax squared plus bx plus c. In this-- with equations written in this form, the values of a, b, and c can be used in something called the quadratic formula, which is used to find solutions for x in a quadratic equation.
In vertex form, y is equal to a times x minus h squared plus k, the values that we have for h and k are the vertex of the graph of the quadratic equation. And in factored form, y is equal to a times x minus x1 times x minus x2. x1 and x2 are the x-intercepts, the values of the x-intercepts for the graph of the quadratic equation.
So let's look at some examples of the graphs of quadratic equations. I have a quadratic equation y is equal to x squared minus 4, which is the graph in red-- corresponds to the graph in red. And I have the equation y equals negative x squared plus 8x minus 16, which corresponds to the graph in blue.
So both of these equations create graphs. And the shapes of these graphs we call parabolas. So the shape of a quadratic equation we call a parabola. And a parabola has a maximum or a minimum point. So for this parabola, we see that the maximum point is here. And for this parabola, the minimum point is here.
The maximum or the minimum point of a parabola is called the vertex. And the vertex, whether a maximum or minimum, is going to lie on a line of symmetry through the middle of the parabola. So this vertex is right on the line of symmetry for the parabola. Similarly, with this vertex.
And when you're looking at the equation of a quadratic, the leading term of that quadratic equation, whether it's positive or negative will tell you if the graph is going to open up or down. So if the leading term is positive, then logically the graph is going to open upwards, so it looks like a U shape. And if the leading term is negative, then the graph will open downward, so it looks like upside down U shape.
So let's look at how to find the solution to a quadratic equation. The solution to a quadratic equation is also sometimes called the zeroes or the roots of the equation. And there's three different methods for solving-- graphing, factoring, and the quadratic formula. Factoring and the quadratic formula are usually the most common.
But for the graphing method, you simply graph the equation. And then you look for the x-intercepts or intercept on the graph, and those values at the x-intercepts are at the solution to your equation.
For the factoring method, you start by writing your quadratic equation in this form. And then you solve two equations-- ax minus p equals 0, and bx plus q equals 0. And the values, or value, of x that you get are your solution-- is the solution to your equation.
And finally, with the quadratic formula, which is useful when you can't write the equation in this factored form, you start by writing your equation in standard form, and then you use this quadratic formula to find the value or values of x. And that is your solution to the equation.
So let's go over a key points from today. Quadratic equations have constant second differences. The graph of a quadratic equation is called a parabola. Parabola have either a minimum or a maximum point, which is called the vertex. The solution to a quadratic equation can also be called a root or a 0.
The solution represents the x-intercepts of the parabola on a graph. And a quadratic equation can be solved algebraically by substituting 0 for y in the equation, and solving for x. So I hope that these key points and examples helped you understand a little bit more about quadratic equations. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.
the shape of a quadratic equation on a graph; it is symmetric at the vertex
a second degree polynomial, with an x-squared term as its highest degree term
the maximum or minimum point of a parabola located on the axis of symmetry