Problem 17 - Calculus with Concepts in Calculus

Question

Sketch the graph. List the intercepts and describe the symmetry (if any) of the graph. $$ y=\frac{1}{3} x $$

Step-by-Step Solution

Answer

The graph is a straight line through (0,0) with symmetry about the origin.

Step 1: Identify the Type of Function

The given function is a linear function, which means its graph will be a straight line. The equation is in the form \( y = mx + b \) where \( m = \frac{1}{3} \) is the slope and \( b = 0 \) is the y-intercept.

Step 2: Find the Intercepts

To find the y-intercept, set \( x = 0 \). The equation becomes \( y = \frac{1}{3} \times 0 = 0 \), thus the y-intercept is at the point \( (0,0) \).For the x-intercept, set \( y = 0 \). The equation becomes \( 0 = \frac{1}{3} x \). Solving for \( x \) gives \( x = 0 \), thus the x-intercept is also at the point \( (0,0) \).

Step 3: Describe the Symmetry

The function \( y = \frac{1}{3}x \) is symmetric with respect to the origin. This is because for every point \( (x, y) \) on the line, the point \( (-x, -y) \) is also on the line. This is a characteristic of odd functions and linear functions that pass through the origin.

Step 4: Sketch the Graph

To graph the line, plot the intercept point \( (0,0) \). From this point, use the slope \( \frac{1}{3} \) to find another point. From \( (0,0) \), go up 1 unit and right 3 units to plot the point \( (3,1) \). Draw a straight line through these points extending in both directions.

Key Concepts

Linear Functions

Linear functions are an essential part of algebra and graphing. They are functions that graph as straight lines. A linear function has the general form:

• \( y = mx + b \)

In this formula:

• \( m \) is the slope, which determines the line's steepness.
• \( b \) is the y-intercept, indicating where the line crosses the y-axis.

For instance, in the function \( y = \frac{1}{3}x \), the slope \( m \) is \( \frac{1}{3} \) and the y-intercept \( b \) is 0. This means the graph is a line that goes through the origin and rises 1 unit for every 3 units it moves to the right. Understanding linear functions helps in predicting trends and establishing relationships between variables. They are often used in various real-world applications such as finance, physics, and social sciences.

Intercepts

Intercepts are crucial for sketching graphs effectively. They tell us where the graph intersects the axes. There are two types of intercepts:

• **Y-intercept**: The point where the graph crosses the y-axis. To find it, set \( x = 0 \) in the equation. In \( y = \frac{1}{3}x \), substituting \( x = 0 \) results in \( y = 0 \). Thus, the y-intercept is \( (0,0) \).
• **X-intercept**: The point where the graph crosses the x-axis. To find this, set \( y = 0 \) in the equation. So, solving \( 0 = \frac{1}{3}x \) gives \( x = 0 \). Hence, the x-intercept is also \( (0,0) \).

For the function \( y = \frac{1}{3}x \), the graph passes through the origin \( (0,0) \). With linear equations, often the intercepts provide an easy way to sketch the graph quickly, as they mark the points of intersection with the axes.

Symmetry

Symmetry in graphs indicates that one part of the graph is a mirror image of another part. For linear functions, especially those which pass through the origin, the concept of origin symmetry is common. A function is symmetric with respect to the origin if for every point \((x, y)\), the point \((-x, -y)\) is also on the graph. This is typical for odd functions. For our function \( y = \frac{1}{3}x \):

• If you take a point like \((3, 1)\), its symmetric point \((-3, -1)\) is also on the line.

This type of symmetry makes the graph aesthetically balanced around the origin. Understanding symmetry aids in predicting the graph's shape and behavior without having to compute every point, making it a valuable tool in graph sketching.