Vertex of a Parabola
The vertex of a parabola is the high point or low point of the graph. The method you use to find the vertex will depend on the form in which the function is given.

You will want to use one strategy when the function is given in vertex form . The reason this is called the vertex form is because the vertex is at the point (h, k). Notice that the value of h is the opposite of what it is in the function, but the value of k has the same sign. So when a function is already presented in this form, the vertex is found simply by looking at the numbers in the function.

You will want to use a different strategy when the function is given in standard form . You can either use a formula or manipulate the function into vertex form. All strategies will be shown in the examples below.

Examples
 #1: Find the vertex of . What is your answer?
 #2: Find the vertex of . This function is in vertex form. The coefficient of 3 will have an effect on the graph, but not on the vertex. To learn more about how a coefficient effects the graph of a parabola, click here to go to the lesson on translating parabolas. What is your answer?
 #3: Find the vertex of What is your answer?
When a function is in standard form rather than vertex form , we cannot simply look at the function and find the vertex of (h, k). There are two ways to approach this problem.
Using a formula: The x-value of the vertex can be found using the formula . Once you know the x-value of the vertex, you can substitute it in the original function to find the y-value.

Completing the square: In the examples below, the procedure will be used with some explanation. To learn more about using the method of completing the square, click here for a detailed lesson.

Examples
 #4a: Find the vertex of using the Formula Method. Find the value of b and a. b = 6 and a = 1 Substitute into the formula to find the x-coordinate of the vertex. Substitute the value of -3 into the function to get the y-value of the vertex. What is your answer?
 #4b: Find the vertex of using the Completing the Square Method. Start with the original function . Take half of 6, square it, then add that to both sides. Rearrange the equation so that it is in vertex form. What is your answer?
 #5a: Find the vertex of by using the Formula Method. Find the value of b and a. b = 6 and a = -3 Substitute into the formula to find the x-coordinate of the vertex. Substitute this value into the function and calculate the y-value of the vertex. What is your answer?
 #5b: Find the vertex of by using the Completing the Square Method. Start by factoring part of the original function Take half of 2, square it, multiply by -3, then add that to both sides. Rearrange the equation so that it is in vertex form. What is your answer?

S Taylor

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