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Let's look at our objectives for today. We'll start by reviewing parabolas. We'll then look at how to calculate the vertex of a parabola. We'll see how to determine the x- and y-intercepts of a parabola. And finally, we'll do some examples identifying a graph from an equation of a parabola.
Let's start by reviewing parabolas. The graph of a quadratic equation is called the parabola, which can either open upwards or downwards. Parabolas have a vertex, which represents a minimum or maximum point on the graph depending on the direction of the parabola.
Quadratic equations in the form below, y equals ax squared plus bx plus c, can give us information about the shape and vertex of the parabola. If the value of the a-coefficient in the equation is positive, the parabola opens upwards and the vertex represents a minimum point. If the value of a coefficient in the equation is negative, the parabola opens downwards and the vertex represents a maximum point.
Now let's review the vertex of a parabola. Equations in the form below, y equals ax squared plus bx plus c, can also help us determine the coordinate of the vertex of the parabola. The values of a and b, and the formula below, x equals negative b over 2 times a, are used to find the x-coordinate of the vertex of the parabola. Once we have the x-coordinate of the vertex of the parabola, we can then substitute this value into the quadratic equation to find the y-coordinate of the vertex.
Now let's look at how to find the intercepts of a parabola. The y-intercept is the point at which the graph intersects the y-axis. On the y-axis the value of x is always 0. Therefore, we can determine the y-intercept by substituting 0 for x in the equation and solving for y. In the general equation below, y equals ax squared plus bx plus c, when x equals 0 the x squared term will equal 0, and the x term will equal 0. Therefore, the y-intercept of any quadratic equation is given by the constant term c.
The x-intercepts are the points at which the graph intercepts the x-axis. On the x-axis, the value of y is always 0. Therefore, we can determine the x-intercepts by substituting 0 for y in the equation and solving for x. Therefore, finding the x-intercepts of any parabola is the same as solving the quadratic equation, since the solutions to a quadratic equation are found by setting the equation equal to 0. The x-intercepts of a parabola are also known as the roots, 0's, or solutions of the quadratic equation.
Now let's look at how to identify a graph from a quadratic equation. Given the quadratic equation below, y equals x squared minus 2x minus 3, which of these four graphs represents the equation? We can use information about the sign of the coefficient a, the vertex, and the x and y-intercepts to eliminate possible graphs.
First, we see that the value of a is 1, which is positive. Therefore, we know that the graph should be facing upwards, so we can eliminate the last graph. Next, we see that the value of c is negative 3, which means the y-intercept of the graph is negative 3. Therefore, we can eliminate the first graph.
Next, we can use the formula for the x-coordinate of the vertex to find that x value of the vertex. Substituting our values for b and a, gives us negative, negative 2 over 2 times 1. This simplifies to positive 2 over positive 2, which equals 1. Therefore, the x-coordinate of our vertex is a positive 1. And we can eliminate the second graph, which has an x-coordinate of negative 1.
Therefore, the graph which matches our equation is the third graph here. The x-intercepts are at negative 1, 0 and 3, 0. These are the roots or 0's of the equation, or the x values, that make y equal to 0.
Let's go over our important points from today. Make sure you get these in your notes so you can refer to them later. The y-intercept of a graph is when the value of x is equal to 0. The y-intercept of any quadratic equation is given by the constant term c.
The x-intercept of a graph is when the value of y is equal to 0. Therefore, finding the x-intercept of a parabola is the same as solving the quadratic equation, since the solutions to a quadratic equation are found by setting the equation equal to 0. The x-intercepts of a parabola are also known as the roots, 0's, or solutions to a quadratic equation.
So I hope that these important points and examples helped you understand a little bit more about identifying the graph of a quadratic equation. Keep using your notes, and keep on practicing, and soon you'll be a pro. Thanks for watching.
00:00 – 00:37 Introduction
00:38 – 01:25 Parabolas
01:26 – 02:04 Calculating the Vertex of a Parabola
02:05 – 03:26 Determining the x and y Intercepts of a Parabola
03:27 – 05:06 Identifying a Graph from an Equation
05:07 – 06:07 Important to Remember (Recap)
x = -b/2a