[MUSIC PLAYING] Let's go over our objectives for today. We'll start by looking at what is a parabola. We'll then talk about the vertex of a parabola. We'll look at the x-intercepts of a parabola. And finally, we'll look at how to find the solution to a quadratic equation by looking at a graph.
Now, let's talk about parabolas. A parabola is the shape of the graph of a quadratic equation. Parabolas have a general U shape to them, either opening up or opening down. In the general quadratic equation below, the coefficient a determines the upward or downward shape of the parabola.
When the coefficient a is positive, the parabola will open upward. And when the coefficient a is negative, the parabola will open downward. So even though we don't know the equation to these graphs, we can see if the value of a is positive or negative. Graphing parabolas can model data such as objects in motion under gravity, area, and optimization.
Now, let's talk about the vertex of a parabola. Every parabola has either a low point or a high point on the graph called the "vertex." In this graph, the parabola has a low point or minimum point. The minimum point has the lowest y value on the parabola. In this graph, the parabola has a high point or a maximum point and the maximum point has the highest y value on the parabola.
Looking at the equations of a quadratic graphs, if the a coefficient of a quadratic equation is positive, the vertex is a minimum point. And if the a coefficient is negative, the vertex is a maximum point.
Let's talk about the x-intercepts of a parabola. The x-intercept of a graph is a point where that graph intersects the x-axis and when y equals 0. The y-intercept of a graph is a point where the graph intersects the y-axis and when x is 0.
On a parabola, the x-intercepts are the x values that make y equal to 0 and they also correspond to the solutions of the quadratic equation. So in this example, the quadratic equation x squared minus 8x plus 15 equals 0 can be solved by a graphing the equation y equals x squared minus 8x plus 15 and identifying the x-intercepts of the graph. So here, our x-intercepts are 3, 0 and 5, 0. And so the solutions to the quadratic equation are x equals 3 and x equals 5.
Let's look a little more at the solutions to a quadratic equation by looking at the graph. Here's another example of a parabola, this time with the equation y equals negative 1 1/2 x squared minus 2x. The vertex of this graph is a maximum point and is at the point negative 2, 2 on the graph. The x-intercepts of the graph are here, at the points negative 4, 0 and 0, 0. Again, the x-intercepts are the solution to the quadratic equation, negative 1 1/2 x squared minus 2x equals 0. So the solutions to this equation are x equals negative 4 and x equals 0.
Let's go over our key points from today. Make sure you get these in your notes so you can refer to them later. A parabola is the shape of the graph of a quadratic equation. Parabolas have a general U shape to them, either opening up or opening down.
In the general quadratic equation, the coefficient a determines the upward or downward shape of the graph. If the coefficient of a quadratic equation is positive, the vertex is a minimum point. And if the a coefficient is negative, the vertex is a maximum point. The x-intercepts of a parabola are the x values that make y equals 0 and also correspond to the solutions of the quadratic equation.
So I hope that these important points and examples helped you understand a little bit more about identifying points on a parabola. Keep using your notes, and keep on practicing. And soon, you'll be a pro. Thanks for watching.
The maximum or minimum point of a parabola.