how to find the vertex of a parabola
How to Find the Vertex of a Parabola
To find the vertex of a parabola, you can use either the vertex form of a quadratic equation or complete the square. The vertex form of a quadratic equation is given by:
f(x) = a(x - h)^2 + k
where ( (h, k) ) represents the coordinates of the vertex. Here’s how to find the vertex using each method:
1. Vertex Form:
If the equation of the parabola is given in vertex form ( f(x) = a(x - h)^2 + k ), then the vertex is simply ( (h, k) ). The value of ( h ) tells you the horizontal shift of the parabola, and ( k ) tells you the vertical shift.
For example, if you have the equation f(x) = 2(x - 3)^2 + 5 , the vertex is at ( (3, 5) ).
2. Completing the Square:
If the equation of the parabola is given in standard form f(x) = ax^2 + bx + c, you can find the vertex by completing the square.
Step 1: Rewrite the equation in the form ( f(x) = a(x - h)^2 + k ).
Step 2: Identify the values of ( h ) and ( k ) to find the vertex ( (h, k) ).
Let’s walk through an example:
Given the quadratic equation f(x) = x^2 - 4x + 3 , we want to find the vertex.
Step 1: Rewrite the equation by completing the square:
f(x) = (x^2 - 4x + \underline{4}) - 4 + 3
Step 2: Identify the values of ( h ) and ( k ):
f(x) = (x - 2)^2 - 1
Comparing this to the vertex form f(x) = a(x - h)^2 + k, we see that ( h = 2 ) and ( k = -1 ). So, the vertex is ( (2, -1) ).
These are the two primary methods used to find the vertex of a parabola. Whether you use the vertex form or complete the square, both methods yield the same result—the coordinates of the vertex.
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