This tutorial covers how to identify points on a parabola, through the definition and discussion of:
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.
- When the coefficient a is positive, the parabola will open upward.
- When the coefficient a is negative, the parabola will open downward.
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.
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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.
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- Vertex (of a Parabola)
- The maximum or minimum point of a parabola
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.
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EXAMPLE
In the graph below, how can you solve the quadratic equation:
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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:
4. Solutions to a Quadratic Equation on a Graph
You can determine the solutions to a quadratic equation by looking at a graph.
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EXAMPLE
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:
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.