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2 Tutorials that teach Identifying Points on Parabola
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Identifying Points on Parabola

Identifying Points on Parabola

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In this lesson, students will learn how to determine the vertex and solutions of a quadratic equation when given its parabolic graph.

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Tutorial
This tutorial covers how to identify points on a parabola, through the definition and discussion of:
  1. Parabolas
  2. Vertex of a Parabola
  3. x-Intercepts of a Parabola
  4. Solutions to a Quadratic Equation on a Graph


1. 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.

y equals a x squared plus b x plus c
  • When the coefficient a is positive, the parabola will open upward.
  • When the coefficient a is negative, the parabola will open downward.

File:1379-parabola1.PNG

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.


2. Vertex of a Parabola

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.

File:1380-parabola2.PNG

Vertex (of a Parabola)
The maximum or minimum point of a parabola
Remember, when looking at the equation of a quadratic graph, if the a coefficient of a quadratic equation is positive, the parabola opens upward and the vertex is a minimum point. If the a coefficient is negative, the parabola opens downward and the vertex is a maximum point.


3. 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 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.

File:1381-parabola3.PNG

For example, in the graph above, how can you solve the following quadratic equation?

x squared minus 8 x plus 15 equals 0

You solve the equation by graphing the equation below and identifying the x-intercepts of the graph.

y equals x squared minus 8 x plus 15

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:

table attributes columnalign left end attributes row cell x equals 3 end cell row cell x equals 5 end cell end table


4. Solutions to a Quadratic Equation on a Graph

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:

y equals negative 1 half x squared minus 2 x

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).

File:1382-parabola4.PNG

Again, the x-intercepts are the solution to this quadratic equation.

negative 1 half x squared minus 2 x equals 0

Therefore, the solutions to this equation are:

table attributes columnalign left end attributes row cell x equals negative 4 end cell row cell x equals 0 end cell end table

Today you learned that a parabola is the shape of the graph of a quadratic equation, and that it has a general U shape, either opening up or opening down. You learned that in the general quadratic equation, the coefficient a determines the upward or downward shape of the graph. You also learned that every parabola has either a low point or a high point on the graph called the vertex, and that if the coefficient of a quadratic equation is positive, the vertex is a minimum point, whereas if the a coefficient is negative, the vertex is a maximum point. Lastly, you learned that the x-intercepts of a parabola are the x-values that make y equal 0 and also correspond to the solutions of the quadratic equation on a graph.

Source: This work is adapted from Sophia author Colleen Atakpu.

TERMS TO KNOW
  • Vertex (of a Parabola)

    The maximum or minimum point of a parabola.