Problem 19 - Algebra 2 and Trigonometry

Question

In \(13-20\) : a. Graph each function. b. Is the function a direct variation? \(c\) . Is the function one-to-one? \(\frac{y}{x}=2\)

Step-by-Step Solution

Answer

The function is \( y = 2x \), a direct variation and is one-to-one.

Step 1: Identify the Function

The given equation is \( \frac{y}{x} = 2 \). This can be rewritten as \( y = 2x \). This function is a linear equation with a slope of 2 and a y-intercept of 0.

Step 2: Graph the Function

To graph the function \( y = 2x \), create a table of values. For example, if \( x = -2 \), then \( y = 2(-2) = -4 \). If \( x = 0 \), then \( y = 2(0) = 0 \). If \( x = 2 \), then \( y = 2(2) = 4 \). Plot these points on a coordinate plane and draw a line through them. The line will pass through the origin and rise at a rate of 2 units vertically for each unit moved horizontally.

Step 3: Determine if the Function is a Direct Variation

A function is a direct variation if it can be expressed in the form \( y = kx \), where \( k eq 0 \). The function \( y = 2x \) is in this form with \( k = 2 \). Therefore, the function is a direct variation.

Step 4: Check if the Function is One-to-One

A function is one-to-one if every \( y \) value is uniquely paired with one \( x \) value. For the function \( y = 2x \), each \( y \) is uniquely determined by \( x \), as no two different \( x \) values produce the same \( y \). Therefore, \( y = 2x \) is a one-to-one function.

Key Concepts

Direct Variation

A direct variation is a specific type of linear function wherein the relationship between the dependent and independent variable is straightforward and proportional. This type of function can be expressed in the form \( y = kx \), where \( k \) is the constant of variation and \( k eq 0 \). The constant \( k \) determines how steep or flat the line will graphically appear. It acts as the slope of the function. In a direct variation, if the independent variable \( x \) changes, the dependent variable \( y \) changes in a consistent manner, multiplied by the constant \( k \). For instance:

• If \( x \) doubles, \( y \) will also double as it is directly proportional.
• If \( x \) is halved, \( y \) will similarly be halved.

In this exercise, the function \( y = 2x \) exemplifies direct variation perfectly with \( k = 2 \). This indicates that for each step increase in \( x \), \( y \) will increase by twice that amount. Graphically, this translates to a line that passes through the origin (0,0). The origin is always a key point in direct variation, emphasizing the proportional and non-shifting nature of these relationships.

One-to-One Functions

A one-to-one function ensures a unique mapping between inputs and outputs in a mathematical relation. This means each element of the domain (input \( x \)) pairs with a unique element of the range (output \( y \)), and vice versa. Put simply,

• No two different \( x \) values can result in the same \( y \) value.
• Each \( y \) corresponds to only one \( x \).

In the context of the function \( y = 2x \), it is one-to-one. For every distinct value of \( x \), there is a unique outcome for \( y \). Mathematically, this can be proven using the horizontal line test. If a horizontal line crosses the graph of the function at more than one point, the function is not one-to-one. The function \( y = 2x \) passes this test as any horizontal line touches the plot at most once, confirming its one-to-one nature. This characteristic allows these functions to be invertible, meaning the function can be reversed without ambiguity.

Graphing Linear Equations

Graphing linear equations is a foundational concept in understanding linear functions. A linear equation represents a straight-line graph, characterized by its slope and y-intercept. The standard form of a linear equation is \( y = mx + b \), where:

• \( m \) represents the slope—it indicates the steepness and direction of the line.
• \( b \) indicates the y-intercept—where the line crosses the y-axis.

For the function \( y = 2x \), the graph will be a line with:
- A slope \( m = 2 \), showing that the line ascends two units vertically for every unit it moves horizontally.
- A y-intercept \( b = 0 \), indicating that the line passes through the origin (0,0).
To graph this function, start by plotting the y-intercept. Then, use the slope to determine other points by moving up 2 units and over 1 unit repeatedly. Connect the dots to form a continuous line. The resulting graph demonstrates the constant rate of change characteristic of linear functions, depicting a proportional increase of \( y \) with respect to \( x \). This method of visualization makes it easier for students to comprehend changes in related variables efficiently.