In this lesson, students will learn how to determine the vertex and solutions of a quadratic equation when given its parabolic graph.
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.
Even though we don’t know the equation to the graphs above, you can see if the value of a is positive or negative. Graphing parabolas can be used to model the path of objects in motion, solve problems involving area, and solve optimization problems.
Every parabola has either a low point or a high point on the graph called the vertex. In the graph below, the parabola has a low, or minimum, point. The minimum point has the lowest y-value on the parabola. The second parabola below has a high, or maximum, point, which has the highest y-value on the 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 equals 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.
For example, in the graph above, how can you solve the following quadratic equation?
You solve the equation by graphing the equation below and identifying the x-intercepts of the graph.
You can see in the graph above that your x-intercepts are (3, 0) and (5, 0). Therefore, the solutions to the quadratic equation are:
You can determine the solutions to a quadratic equation by looking at a graph. Suppose you have another example of a parabola, with the equation:
The vertex of this graph is a maximum point and is at the point (-2, 2) on the graph. The x-intercepts of the graph are at the points (- 4, 0) and (0, 0).
Again, the x-intercepts are the solution to this quadratic equation.
Therefore, the solutions to this equation are:
Source: This work is adapted from Sophia author Colleen Atakpu.
The maximum or minimum point of a parabola.