Problem 18 - Precalculus with Limits

Question

In Exercises 17-22, use the Vertical Line Test to determine whether \(y\) is a function of \(x\). To print an enlarged copy of the graph, go to the website \(www.mathgraphs.com\). \(y=\frac{1}{4}x^3\)

Step-by-Step Solution

Answer

Since no vertical line intersected the graph more than once, \(y = \frac{1}{4}x^3\) is graph of a function.

Step 1: Understand the vertical line test

The vertical line test is a method to determine if a curve is a graph of a function. If all vertical lines intersect a curve at most once then the curve represents a function. If the line intersects more than once, then the curve doesn't represent a function.

Step 2: Graph the equation

The equation given, \(y = \frac{1}{4}x^3\), can be visualised as a cubic function. Its graph will be a curve that passes through the origin (0,0). On the left side of the y-axis, it falls and on the right side of y-axis, it rises.

Step 3: Apply the vertical line test

Draw a number of vertical lines that cross the graph. Observe where these lines intersect the graph. Upon examining each vertical line, we see that no line intersects the graph more than once. This means that the graph represents a function.

Key Concepts

Cubic Function

A cubic function is a type of polynomial function with the highest degree of three. This means the function is expressed in the form \[ y = ax^3 + bx^2 + cx + d \]where \(a\), \(b\), \(c\), and \(d\) are constants, and \(a eq 0\). These constants determine the shape and position of the cubic curve on a graph.

• The term with \(x^3\) indicates the cubic nature of the graph, which typically means that it will be a curve with one or two "turning" points or bends.
• Cubic functions can either have one real root or three real roots, which determine the number of times the graph will cross the x-axis.
• Because of the odd power of the leading term, cubic functions often have a distinctive "S" shape curve.

In the example given, \(y = \frac{1}{4}x^3\), the absence of \(x^2\), \(x\), and constant \(d\) simplifies this into a basic cubic function that specifically passes through the origin. Studying these properties can help you predict how the curve will behave as \(x\) becomes increasingly positive or negative.

Graph of a Function

When we talk about the graph of a function, we're discussing how you can visualize the function by plotting its points on a coordinate plane. This graph gives you an insight into how the function behaves.A function's graph can reveal many things:

• Intercepts: Points where the graph crosses the x-axis or y-axis.
• Increasing or Decreasing: Sections of the graph that slope upwards or downwards.
• Symmetry: Whether or not the graph is symmetric about the y-axis or any other line.

A function's graph not only helps in visualizing how inputs transform into outputs but is crucial in methods such as the Vertical Line Test, which helps in identifying whether equations truly represent functions visually, as seen in the initial exercise. In the case of \(y = \frac{1}{4}x^3\), the graph has a characteristic smooth bend through the origin, rising to the right and falling to the left.

Mathematical Visualization

Mathematical visualization involves creating a visual image to understand and interpret mathematical concepts better. It is especially useful in exploring and understanding functions and their behaviors. Visualizing a function can provide insights that are not immediately apparent from the equation alone.Here are key points about mathematical visualization:

• It can help identify patterns and relationships among different mathematical ideas.
• It allows for easier comprehension and teaching of complex concepts by translating them into visual formats.
• Tools like graphing calculators or software assist in visualizing more intricate functions easily.

In the case of checking whether a graph is a function, as indicated in the original exercise, visualization is key. By plotting \(y = \frac{1}{4}x^3\), you can directly apply the Vertical Line Test. This visual approach allows you to clearly see if every x-input (depicted by a vertical line) maps to a single y-output. Visualization here gives a straightforward graphical evidence that supports the algebraic understanding of the function's nature.