Problem 16 - College Algebra and Calculus: An Applied Approach

Question

Compare the graph of the quadratic function with the graph of \(y=x^{2}\). $$f(x)=-\frac{1}{4} x^{2}$$

Step-by-Step Solution

Answer

The function \(f(x)=-\frac{1}{4} x^{2}\) is a downward-opening parabola, wider than the basic function \(y=x^{2}\), which is an upward-opening parabola.

Step 1: Identify the basic function and its properties

The function \(y=x^{2}\) represents a basic quadratic function. It is a parabola that opens upwards, whose vertex is at the origin (0,0).

Step 2: Identify the characteristics of the second function

Function \(f(x) = -\frac{1}{4} x^{2}\) is also a quadratic function, but it differs in a few aspects. The minus sign before the fraction indicates that the orientation of the parabola is downward. Also, the coefficient of \(x^{2}\) is \(-\frac{1}{4}\), less than 1, which means the parabola is vertically compressed by a factor of \(\frac{1}{4}\).

Step 3: Compare the two functions

Both graphs are parabolas, but they open in opposite directions. The graph of the basic function \(y=x^{2}\) opens upwards, while the graph of the function \(f(x)=-\frac{1}{4}x^{2}\) opens downwards due to the negative sign. Furthermore, the function \(f(x)\) has a vertical compression by a factor of \(\frac{1}{4}\), which means it is 'wider' or extends more horizontally compared to the graph of \(y=x^{2}\).

Key Concepts

Parabola

A parabola is a U-shaped curve that is commonly found in the graphs of quadratic functions. The most basic form of a quadratic function is given by the equation \(y = x^2\). This function forms a parabola that opens upwards, centered around the vertical axis, and has its vertex at the origin (0,0). Parabolas are symmetric and can open upwards or downwards depending on their coefficients.

• If the coefficient of \(x^2\) is positive, the parabola opens upward, like \(y = x^2\).
• If the coefficient of \(x^2\) is negative, the parabola opens downward, as seen in \(f(x) = -\frac{1}{4} x^2\).

When comparing different parabolas, it is important to focus on their direction, width, and position. These aspects can change how the parabola interacts with the coordinate grid and represent different behaviors in real-world applications.

Vertical Compression

Vertical compression is a transformation that affects the width of the parabola. It occurs when the absolute value of the coefficient before \(x^2\) in the quadratic function is less than 1. In the function \(f(x) = -\frac{1}{4} x^2\), this coefficient is \(-\frac{1}{4}\). This indicates that the parabola is compressed vertically by a factor of \(\frac{1}{4}\), making it wider than the standard parabola formed by \(y = x^2\).

• The term 'compression' might sound like the graph is being squeezed, but in this case, it means the graph extends more horizontally.
• A vertical compression reduces how steep the parabola appears, effectively making it appear flatter.

Vertical transformation can significantly alter the appearance of a quadratic graph, which is helpful to consider when analyzing or sketching graphs for various functions.

Vertex

The vertex of a parabola is the point where it turns; it can be thought of as its highest or lowest point. In the function \(y = x^2\), the vertex is at the origin (0,0). When considering transformations, it is essential to track changes to the vertex since they affect the graph's position.

• A basic transformation that includes a negative sign in front of the quadratic term, such as in \(f(x) = -\frac{1}{4} x^2\), does not affect the vertex position horizontally or vertically.
• In this function, even though the parabola opens downward and is vertically compressed, the vertex remains at (0,0).

Understanding the vertex's role is crucial when modeling real-world problems using quadratic functions, as the vertex often corresponds to maximum or minimum values of the scenario being modeled.