In this lesson, students will learn how to identify the parabola that represents a given quadratic equation.
In review, 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 can provide information about the shape and vertex of the parabola:
Equations in the quadratic form below can also help you determine the coordinate of the vertex of the parabola.
The values of a and b, circled above, and the vertex formula below are used to find the x-coordinate of the vertex of the parabola.
Once you have the x-coordinate of the vertex of the parabola, you can then substitute this value into the quadratic equation to find the y-coordinate of the vertex.
Given what you’ve learned so far, how can you find the intercepts of a parabola? The y-intercept is the point at which the graph intersects the y-axis, and on the y-axis the value of x is always 0. Therefore, you can determine the y-intercept by substituting 0 for x in the equation and solving for y. In the general equation below, when x equals 0, the x^2 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, you can determine the x-intercepts by substituting 0 for y in the equation and solving for x.
Given a quadratic equation, you can use information about the sign of the coefficient a, the vertex, and the x- and y-intercepts to identify its corresponding graph. Given the quadratic equation below, which of these four graphs represents the equation?
First, you can see that the value of a is 1, which is positive. Therefore, you know that the graph should be facing upwards, so you can eliminate the last graph.
Also, you can see that the value of c is -3, which means that the y-intercept of the graph is -3. Therefore, you can eliminate the first graph.
Next, you can use the formula for the x-coordinate of the vertex (the vertex formula) to find the x-value of the vertex. Substitute your values for b and a and simplify:
Since your solution is x equals 1, then you know that the x-coordinate of your vertex is a positive 1. Therefore, you can eliminate the second graph, which has an x-coordinate of -1. This means that the remaining graph, the third graph, matches your equation. The x-intercepts are at (-1, 0) and (3, 0). These are the roots or 0s of the equation, or the x-values, that make y equal to 0.
Source: This work is adapted from Sophia author Colleen Atakpu.