Understanding the Vertex Form of a Quadratic Function for Quick Analysis and Insights

vertex form of a quadratic function

The vertex form of a quadratic function is a way to express a quadratic equation in the form:

f(x) = a(x – h)^2 + k

In this form, “a” represents the coefficient of the quadratic term, “(x – h)^2” represents the squared term, and (h, k) represents the coordinates of the vertex of the parabola

The vertex form of a quadratic function is a way to express a quadratic equation in the form:

f(x) = a(x – h)^2 + k

In this form, “a” represents the coefficient of the quadratic term, “(x – h)^2” represents the squared term, and (h, k) represents the coordinates of the vertex of the parabola.

The vertex form allows us to easily determine certain properties of the quadratic function. The vertex, which is given by the coordinates (h, k), is the highest or lowest point on the parabola, depending on the value of “a.” If “a” is positive, the parabola opens upwards, and the vertex represents the minimum point. If “a” is negative, the parabola opens downwards, and the vertex represents the maximum point.

The value of “h” determines the horizontal shift of the parabola. If “h” is positive, the graph shifts “h” units to the right, and if “h” is negative, the graph shifts “h” units to the left.

The value of “k” determines the vertical shift of the parabola. If “k” is positive, the graph shifts “k” units upward, and if “k” is negative, the graph shifts “k” units downward.

To convert a quadratic equation into vertex form, you can use the process of completing the square. This involves taking the coefficient of the linear term, dividing it by 2, and squaring it. Then, add and subtract this value within the parentheses, and adjust the rest of the equation accordingly.

For example, let’s say we have the quadratic equation f(x) = 2x^2 + 4x + 3.

To convert it into vertex form, we complete the square as follows:

f(x) = 2(x^2 + 2x) + 3
= 2(x^2 + 2x + 1 – 1) + 3 (add and subtract (2/2)^2 = 1 within the parentheses to complete the square)
= 2[(x + 1)^2 – 1] + 3
= 2(x + 1)^2 – 2 + 3
= 2(x + 1)^2 + 1 (simplifying)

Now, we can see that the quadratic equation is in vertex form f(x) = 2(x + 1)^2 + 1. The vertex is located at the coordinates (-1, 1), and the parabola opens upwards since “a” is positive.

The vertex form of a quadratic function is often useful for quickly identifying important information about the graph, such as the vertex, direction of opening, and shifts.

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