Problem 17 - 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? \(y=\frac{1}{2} x\)

Step-by-Step Solution

Answer

The function is a direct variation and is one-to-one.

Step 1: Identify the Function Type

The function given is \( y = \frac{1}{2}x \). This is a linear function in the form \( y = mx + b \) where \( m = \frac{1}{2} \) and \( b = 0 \). The graph of this function will be a straight line passing through the origin (0,0).

Step 2: Graph the Function

To graph \( y = \frac{1}{2}x \), plot the y-intercept (0,0). Next, use the slope \( \frac{1}{2} \) to find another point. Starting at (0,0), move up 1 unit and right 2 units to reach (2,1). Draw a line through these points, extending in both directions. This line represents the function.

Step 3: Determine if the Function is a Direct Variation

Direct variation is a relationship of the form \( y = kx \), where \( k eq 0 \). The given function \( y = \frac{1}{2}x \) matches this form with \( k = \frac{1}{2} \). Hence, the function is a direct variation.

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

A function is one-to-one if each output (y-value) is paired with exactly one input (x-value). Linear functions of the form \( y = mx \) are one-to-one as long as \( m eq 0 \). Since \( m = \frac{1}{2} \), the function is one-to-one, as it passes the horizontal line test (each horizontal line intersects the graph at most once).

Key Concepts

Direct Variation

The concept of direct variation is quite straightforward. In mathematics, a direct variation relationship can be described by the equation \( y = kx \), where \( k \) is a non-zero constant.
This means that as one variable changes, the other changes at a constant rate.

• For instance, in the function \( y = \frac{1}{2}x \), \( k \) is \( \frac{1}{2} \).
• The line of the graph will always pass through the origin (0,0), which is a hallmark of direct variation.
• These types of functions are useful in many real-life scenarios where one quantity depends directly on another, such as speed and distance.

So, whenever you see a function of the form \( y = kx \) without a constant being added or subtracted, it's a perfect example of direct variation.

One-to-One Functions

Understanding one-to-one functions is key in determining the uniqueness of outputs. A function is one-to-one if every output corresponds with exactly one input.
In other words, no two different x-values have the same y-value.
This is important because one-to-one functions have the property that they can be inverted.

• For linear functions with the form \( y = mx + b \), if \( m eq 0 \), the function is one-to-one.
• The graph must pass the horizontal line test, meaning any horizontal line drawn through the graph hits no more than one point.

In the case of the function \( y = \frac{1}{2}x \), since the slope \( m \) is \( \frac{1}{2} \), which is not zero, this function is indeed one-to-one.

Graphing Linear Equations

Graphing linear equations is an essential skill whether you're just starting with algebra or exploring advanced mathematics. These equations create straight lines when graphed on a coordinate plane.
The form \( y = mx + b \) is called the slope-intercept form, where \( m \) is the slope, and \( b \) is the y-intercept.

• The slope \( m \) tells us how steep the line is. In \( y = \frac{1}{2}x \), the slope is \( \frac{1}{2} \), indicating an upward incline that rises 1 unit for every 2 units it moves right.
• The y-intercept \( b = 0 \) in this equation tells us the line passes through the origin (0,0).
• To graph this equation, you start at the y-intercept and use the slope to determine the direction and slant of the line.
• For example, from (0,0), you would move up 1 unit and over 2 units to plot the next point.

By connecting these points with a straight line, you can visualize the behavior and characteristics of the linear function.