Problem 54 - Algebra 2

Question

Explain why \(f(x)=\frac{x+2}{2}\) is a linear function.

Step-by-Step Solution

Answer

The function \(f(x) = \frac{x+2}{2}\) is linear because it can be expressed as \(f(x) = \frac{1}{2}x + 1\), a linear equation in the form \(mx + c\).

Step 1: Identify the basic form of a linear function.

A linear function can be represented in the standard form of a line, which is given by: \(f(x) = mx + c\). Here, \(m\) is the slope and \(c\) is the y-intercept, both constants.

Step 2: Rearrange the function to match the linear form.

The function given is \(f(x)=\frac{x+2}{2}\). Simplify this by rewriting it as: \(f(x) = \frac{x}{2} + 1\). This expression resembles the linear form \(f(x) = mx + c\) with \(m = \frac{1}{2}\) and \(c = 1\).

Step 3: Explain each component of the linear form.

In \(f(x) = \frac{x}{2} + 1\), the term \(\frac{x}{2}\) is the variable part and is responsible for creating a straight line due to its linear dependence on \(x\). The constant term \(1\) represents the y-intercept, which shifts the line vertically on a graph.

Step 4: Conclude why \(f(x)\) is linear.

Since \(f(x) = \frac{x}{2} + 1\) follows the form \(f(x) = mx + c\), where both \(m\) and \(c\) are constants, it satisfies all conditions of a linear function. This ensures that its graph is a straight line with a slope of \(\frac{1}{2}\) and a y-intercept of \(1\).

Key Concepts

Slope

The slope of a linear function is a crucial component that determines the direction and steepness of the line on the graph. In a linear equation of the form \( y = mx + c \), the slope \( m \) is the coefficient of the variable \( x \). It indicates how much the value of \( y \) will change for a unit change in \( x \).
For example, if the slope \( m = \frac{1}{2} \), it means that for every 1 unit increase in \( x \), the value of \( y \) increases by \( \frac{1}{2} \) units. This positive slope indicates an upward trend as we move from left to right on the graph.

• A positive slope results in an upward slanting line.
• A negative slope results in a downward slanting line.
• A zero slope means the line is horizontal.
• An undefined slope corresponds to a vertical line.

Understanding the slope helps us predict the behavior of the function without graphing it.

y-intercept

The y-intercept is another core element of a linear function. It is the point where the line crosses the y-axis, represented by \( c \) in the equation \( y = mx + c \). In our equation, \( f(x) = \frac{x}{2} + 1 \), the y-intercept is \( 1 \). This implies that when \( x = 0 \), \( y \) will equal \( 1 \).

• The y-intercept provides a starting point for the line on a graph.
• It tells us the value of the function when there are no contributions from the \( x \) variable.
• This makes it a vital component for sketching the graph accurately.

The y-intercept gives a quick insight into where the line sits relative to the origin.

Graph of a function

When graphing a linear function, the two main components to consider are the slope and y-intercept. These two elements form the foundation for sketching the graph effectively.
Start by plotting the y-intercept on the y-axis. From the y-intercept, use the slope to determine the direction of the line: move over by 1 unit on the x-axis, and then move up or down depending on the value of the slope.
For the function \( f(x) = \frac{x}{2} + 1 \):

• Start at \( (0, 1) \) because the y-intercept is 1.
• From there, move right 1 unit (because of the '1' in the slope fraction's denominator) and up \( \frac{1}{2} \) unit (the numerator) to mark the next point.
• Draw a straight line through these points to extend the function across the graph.

Following these steps results in a clear visual representation of the linear function. The graph will show a straight line, which confirms the properties discussed regarding slope and y-intercept, helping to solidify understanding of the function's behavior.