Problem 17 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describe the graph of the function and identify the vertex. Use a graphing utility to verify your results. \(f(x)=20-x^{2}\)

Step-by-Step Solution

Answer

The described graph is an inverted parabola that opens downward. The vertex of the function \(f(x)=20-x^{2}\) is at point \((0, 20)\).

Step 1: Identify The Form

The given function \(f(x)=20-x^{2}\) can be written in the form \(f(x)= a - (x-h)^{2} + k\), where \(a\), \(h\), and \(k\) are real numbers. Here \(a = -1\), \(h = 0\), and \(k = 20\). It's important to recognize that the coefficient of \(x^{2}\) is negative, which indicates this function describes an inverted parabola and opens downward.

Step 2: Identify the Vertex

The vertex of the function \(f(x)=20-x^{2}\) is the maximum point on the function (since the function opens downward). The vertex for a function in the form \(f(x) = a - (x-h)^{2} + k\) is located at \((h, k)\). Therefore, for this function, the vertex is at \((0, 20)\).

Step 3: Verification using a Graphing Utility

Plug the function \(f(x)=20-x^{2}\) into a graphing calculator or utility. The resulting shape will be an inverted parabola, opening downward, with the vertex at the point \((0, 20)\). This confirms the earlier findings.

Key Concepts

Vertex of a Parabola

Understanding the vertex of a parabola is crucial when studying quadratic functions. In the graph of a quadratic function, the vertex is the point where the parabola changes direction, which can be either the highest or lowest point on the graph, depending on the parabola's orientation.

For the given function
\(f(x) = 20 - x^2\)

, the vertex is found by rewriting the equation in the vertex form, which is
\(f(x) = a(x - h)^2 + k\)

where \((h, k)\) is the vertex of the parabola. By comparison, we find
\(h = 0\)
and
\(k = 20\)

. Therefore, the vertex for this particular function is \((0, 20)\). This point represents the maximum value that the function attains, and it is a pivotal concept in understanding the graph's shape and properties.

Parabola Opening Downward

A parabola that opens downward has a concave shape like an upside-down 'U' and is characteristic of quadratic functions with a negative leading coefficient. In our function
\(f(x) = 20 - x^2\)

, the coefficient of
\(x^2\) is
\(-1\)
, which is less than zero. This negative coefficient is the hallmark of a parabola opening downwards.

When graphing such functions, the vertex becomes the highest point, also known as the maximum. The arms of the parabola extend downwards from the vertex, widening as they go further from it. Knowing that a parabola opens downward helps in various applications, such as determining the maximum height of a projectile or solving optimization problems.

Graphing Utility Verification

When studying quadratic functions, verifying your sketches with a graphing utility not only solidifies your understanding but also ensures accuracy. After manually determining that the vertex of the function
\(f(x) = 20 - x^2\)

is
\((0, 20)\)
and that the parabola opens downward, you can input the function into a graphing calculator or online graphing tool to visualize the curve.

A proper graphing utility will show a parabola dipping downwards, confirming that the vertex is the peak of this curve. It's an excellent way to check your work and develop intuition for the behavior of quadratic functions. Using graphing utilities as part of your learning process can provide instantaneous feedback and reinforce key concepts like vertices and the direction of opening for parabolas.