Problem 14 - 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? \(y=6-x\)

Step-by-Step Solution

Answer

The function \( y = 6-x \) is a linear function that is not a direct variation, but it is one-to-one.

Step 1: Identify the type of function

The given function is in the form of a linear equation, specifically, \( y = 6 - x \). This can be rewritten as \( y = -x + 6 \), which is a linear function with a slope of \(-1\) and a y-intercept of \(6\).

Step 2: Graph the function

To graph \( y = 6 - x \), start by plotting the y-intercept at \((0,6)\). Then, use the slope of \(-1\) to find another point. From \((0,6)\), move down 1 unit and right 1 unit to reach \((1,5)\). Connect these points with a straight line, extending in both directions.

Step 3: Determine if the function is a direct variation

A direct variation function is in the form \( y = kx \), where \( k \) is a constant. The function \( y = 6 - x \) can be rearranged to \( y = -x + 6 \) and clearly shows that it is not in the form \( y = kx \). Therefore, it is not a direct variation.

Step 4: Check if the function is one-to-one

A function is one-to-one if it passes the Horizontal Line Test, meaning any horizontal line intersects the graph at most once. Since \( y = 6 - x \) is a linear function with non-horizontal slope, it will pass any horizontal line only once. Therefore, the function is one-to-one.

Key Concepts

Direct Variation

In mathematics, direct variation is a relationship between two variables where one is a constant multiple of the other. This type of function can be represented by the equation \( y = kx \), with \( k \) being the constant of variation. It's essential to identify whether a function is a direct variation, as it reveals a proportional relationship.
A direct variation has the following characteristics:

• The graph is a straight line passing through the origin (\(0, 0\)).
• The formula takes the form \( y = kx \), where \( k \) is not zero.
• As one variable increases, the other also increases or decreases consistently based on \( k \).

In the case of the function \( y = 6 - x \), this is not a direct variation. The equation can be rearranged to \( y = -x + 6 \), which does not fit the \( y = kx \) format because of the constant term \( 6 \). This constant prevents the graph from passing through the origin. Hence, the function does not demonstrate direct variation.

One-to-One Functions

One-to-one functions are unique because they exhibit a special relationship where each input has a distinct output, and vice versa. This means no two different inputs will ever produce the same output.
To identify a one-to-one function, you can use the horizontal line test. This test involves drawing horizontal lines across the graph of the function. If any horizontal line intersects the graph more than once, the function is not one-to-one.

• Distinct inputs lead to distinct outputs.
• Passes the horizontal line test: any horizontal line cuts the graph only once.

The function \( y = 6 - x \) is considered one-to-one due to its linear nature and non-horizontal slope. When using the horizontal line test on this function, each horizontal line crosses the line only once. Therefore, \( y = 6 - x \) passes the criteria and is indeed a one-to-one function.

Linear Equations

Linear equations are fundamental in algebra and can be represented by straight lines on a graph. These equations are characterized by the highest exponent of their variable being one, and they often take the form \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.
Some key points about linear equations include:

• The graph of a linear equation is always a straight line.
• The slope \( m \) indicates the steepness and direction of the line (a positive slope means the line rises, while a negative slope means it falls).
• The y-intercept \( b \) specifies where the line crosses the y-axis.

For the specific function \( y = 6 - x \), which can be rewritten as \( y = -x + 6 \), we identify the slope \(-1\) and the y-intercept \(6\). This information allows us to easily graph the line by starting at \( (0, 6) \) on the y-axis and applying the slope to determine other points on the line. Linear equations like this one are straightforward to work with because of their predictable and consistent nature.