Lesson 6.3: Reciprocal and Other Simple Non-Linear Graphs

Introduction

In this lesson, we will explore some fundamental concepts of reciprocal and cubic functions in mathematics. We will understand how functions can be represented graphically and how to identify their features and behaviors. The main objective of this lesson is to:

• Examine the graph of the function $y = \frac{1}{x}$ and understand the behavior of the curve approaching the axes.
• Explore the shapes of simple cubic and reciprocal graphs and recognize their characteristics.
• Learn how to match a graph to its corresponding equation by observing key features.
• Sketch the graph of a reciprocal function accurately.
• Understand the shapes of simple cubic and reciprocal functions.

By the end of this lesson, students will be equipped with the necessary tools to identify and analyze these important mathematical functions.

Understanding the Reciprocal Function

1. The Graph of $y = \frac{1}{x}$

The reciprocal function is defined as follows:

$$

$ y = \frac{1}{x}$

$$

This function relates a number $x$ to its reciprocal $y$. To understand the graph, we need to evaluate the function at different values of $x$:

• When $x = 1$, $y = \frac{1}{1} = 1$.
• When $x = 2$, $y = \frac{1}{2} = 0.5$.
• When $x = -1$, $y = \frac{1}{-1} = -1$.
• When $x = -2$, $y = \frac{1}{-2} = -0.5$.
• As $x$ approaches $0$ from the positive side, $y$ approaches $+\infty$.
• As $x$ approaches $0$ from the negative side, $y$ approaches $-\infty$.

Now, if we plot these points, we find that the graph of $y = \frac{1}{x}$ has two distinct branches:

• The right branch is in Quadrant I where both $x$ and $y$ are positive.
• The left branch is in Quadrant III where both $x$ and $y$ are negative.

Key Features of the Graph:

1. Vertical Asymptote at $x = 0$: The graph approaches the y-axis but never touches or crosses it.
2. Horizontal Asymptote at $y = 0$: The graph approaches the x-axis but never intersects it.
3. Domain: All real numbers except $x = 0$. This is written as $(-\infty, 0) \cup (0, \infty)$.
4. Range: All real numbers except $y = 0$. This is written as $(-\infty, 0) \cup (0, \infty)$.

Worked Example 1

Graph the function $y = \frac{1}{x}$ on the interval $[-4, 4]$.

1. Evaluate $y$ at specific points: $x = -4, -2, -1, -0.5, 0.5, 1, 2, 4$.
2. The calculated $y$ values will be: $(-0.25, -0.5, -1, -2, 2, 1, 0.5, 0.25)$.
3. Plot the points: $(-4, -0.25), (-2, -0.5), (-1, -1), (-0.5, -2), (0.5, 2), (1, 1), (2, 0.5), (4, 0.25)$.
4. Draw the two branches of the graph, approaching the axes as described.

Characteristics of Simple Cubic Functions

2. An Introduction to Cubic Functions

A simple cubic function can be represented in the form:

$$

$ y = x^3$

$$

This function exhibits different properties compared to quadratic functions. Let’s analyze the basic features of the cubic function:

• When $x = -1$, $y = (-1)^3 = -1$.
• When $x = 0$, $y = 0^3 = 0$.
• When $x = 1$, $y = 1^3 = 1$.
• When $x = 2$, $y = 2^3 = 8$.
• When $x = -2$, $y = (-2)^3 = -8$.

Key Features of the Cubic Function:

1. Shape: The graph of $y = x^3$ passes through the origin and has a distinctive S-shape.
2. Inflection Point: The curve changes concavity at the origin $(0, 0)$ where $y = 0$. For $x < 0$, the graph is increasing and concave down, while for $x > 0$, the graph is increasing and concave up.
3. Domain and Range: Both the domain and range are all real numbers, $(-\infty, \infty)$.

Worked Example 2

Graph the function $y = x^3$ on the interval $[-2, 2]$.

1. Evaluate $y$ at the points: $-2, -1, 0, 1, 2$.
2. The corresponding $y$ values will be: $(-8, -1, 0, 1, 8)$.
3. Plot the points: $(-2, -8), (-1, -1), (0, 0), (1, 1), (2, 8)$.
4. Connect the points smoothly, exhibiting the typical cubic curve behavior.

Matching Graphs to Equations

3. Identifying Key Features

To match a graph to its equation:

• Identify intercepts (e.g., $x$ and $y$-intercepts).
• Check the symmetry: A cubic function $y = x^3$ is symmetric about the origin.
• Look for asymptotes which are present in reciprocal functions.

Worked Example 3

Given the following graphs, which corresponds to $y = \frac{1}{x}$, and which corresponds to $y = x^3$?

• Observe there is one graph that approaches both axes and does not intersect them, likely indicating it represents $y = \frac{1}{x}$.
• The other graph shows a smooth curve that passes through the origin and has no bounds, indicative of the cubic graph $y = x^3$.

Sketching General Shapes

4. How to Sketch a Reciprocal Graph

When sketching the graph of $y = \frac{1}{x}$:

1. Draw the coordinate axes.
2. Plot the vertical and horizontal asymptotes.
3. Plot a few key points calculated earlier.
4. Drawing a smooth curve that approaches the asymptotes helps visualize the function accurately.

Conclusion

By understanding the reciprocal function and cubic functions, students is now able to recognize their behavior and characteristics on graphs. Identifying the graph's features such as intercepts, asymptotes, and regions of increase or decrease is crucial for comprehensive learning in mathematics. Through practice, students will gain further insight into applying these concepts.

Study Notes

• The graph of $y = \frac{1}{x}$ has vertical and horizontal asymptotes.
• The function $y = x^3$ exhibits inflection points and provides an S-shape curve.
• Both functions have distinct domain and range characteristics.
• Being able to identify the properties of a graph enables proper analysis and understanding of function behavior.