Question

Write the quadratic function in vertex form. Then identify the vertex. \(h(x)=x^2+20 x+90\)

Short answer

The vertex form of the function is \(h(x) = (x+10)^2 - 10\), and the vertex is \((10,-10)\)

Step-by-step solution

Rewriting Quadratic Function

First, rewrite the quadratic function by grouping the \(x\) terms together and leaving the constant term aside. Like this: \(h(x) = (x^2 +20x) + 90\)

Completing the Square

To complete the square, add and subtract the square of half the coefficient of \(x\) (which is \(20/2 = 10\) in this case) inside the parentheses. This will look like: \(h(x) = [(x^2 + 20x + 10^2) - 10^2] + 90\). Then you simplify and write it as \(h(x) = (x + 10)^2 - 100 + 90\)

Simplifying The Function

Now simplify the equation by combining the constant terms which leads to : \(h(x) = (x + 10)^2 - 10\)

Identifying The Vertex

The vertex of the parabola given by the equation \(h(x) = a(x-h)^2 + k\) is \((-h, k)\). In this case, \(h=-10\) and \(k=-10\), so the vertex of the parabola is \((-(-10), -10) = (10 , -10)\)

Key concepts

Completing the Square

To transform a quadratic equation into vertex form, completing the square is an essential process that allows the equation to reveal its vertex. Essentially, it involves creating a perfect square trinomial from the quadratic term and the linear term, which then can be expressed as the square of a binomial.

The core steps are to first divide the linear coefficient by 2, square the result, and then add and subtract this square within the quadratic function. In the example solution, the linear coefficient is 20, so half of it is 10, and squaring 10 gives us 100. By both adding and subtracting 100, you maintain the function's original value while manipulating it into a form that is beneficial for further simplification. The quadratic function now factors neatly into \( (x + 10)^2 \) with an additional constant that needs to be simplified.

This technique not only prepares the equation for easy identification of the vertex but also aids in graphing the parabola and solving quadratic equations by finding roots.

Vertex of a Parabola

The vertex of a parabola is the 'turning point' where the graph changes direction. In the context of a quadratic function \( ax^2 + bx + c \), the vertex form \( a(x-h)^2 + k \) is particularly useful because it clearly identifies the vertex as the point (h, k).

When given a quadratic equation not initially in vertex form, like the example solution \( h(x) = x^2 + 20x + 90 \), completing the square is a method that aids in rewriting the function such that the vertex coordinates become evident. After completing the square and simplifying, we find that the vertex form of the quadratic equation is \( h(x)=(x+10)^2-10 \) which suggests that the vertex is at \(h = -10\) and \(k = -10\), yielding a vertex point of (10, -10). Knowing the vertex is crucial for graphing and understanding the parabola's properties, such as the axis of symmetry and the maximum or minimum point.

Simplifying Quadratic Equations

Simplifying quadratic equations involves reducing them to their simplest form while retaining the same properties and solutions. Techniques such as combining like terms, factoring, and completing the square assist in this simplification.

The given step-by-step solution demonstrates how to simplify a quadratic equation by combining like terms after completing the square. The original quadratic equation \( h(x)=x^2+20x+90 \) was rearranged and transformed into the vertex form \( h(x)=(x+10)^2-10 \), where constant terms were combined resulting in the simplified version.

Simplifying may lead to an equation that is easier to evaluate for its zeroes, vertex, and line of symmetry. It often provides a clearer view of the quadratic function's graph and allows for straightforward comparison with other quadratic functions, enhancing the analysis of their characteristics and behavior.