Problem 19 - Beginning and Intermediate Algebra

Question

Graph each function. $$f(x)=x-5$$

Step-by-Step Solution

Answer

The graph of the function \(f(x) = x - 5\) is a straight line that passes through the points (0, -5), (1, -4), and (2, -3). The line has a slope of 1 and a y-intercept at -5.

Step 1: Identify the slope and y-intercept of the function

The given function is in the form of \(f(x) = mx + b\), which is the standard form of a linear equation. 'm' represents the slope and 'b' represents the y-intercept. In our given function, \(f(x)=x-5\), we can see that the slope (m) is 1 and the y-intercept (b) is -5.

Step 2: Plot the y-intercept on the coordinate plane

The y-intercept represents the point at which the line intersects the y-axis. Since our y-intercept is -5, we plot a point at the coordinates (0, -5).

Step 3: Use the slope to plot at least two more points on the line

The slope is 1, which represents a rise of 1 and a run of 1. Starting from the y-intercept, we can use the slope to find two more points on the line. For example, we can move one unit up and one unit to the right from the y-intercept to find another point at (1, -4). Similarly, we can move another unit up and one unit to the right to find a third point at (2, -3).

Step 4: Draw the line through the plotted points

Now that we have three points on the line, we can draw a straight line passing through these points on the coordinate plane. This represents the graph of the function \(f(x) = x-5\). By following these steps, we have successfully graphed the linear function \(f(x)=x-5\).

Key Concepts

Linear Equations

Linear equations are a type of equation where the solutions form a straight line when graphed on a coordinate plane. These are among the simplest types of equations due to their consistent slope. To identify a linear equation, you will often see it written as \( y = mx + b \), which explicitly indicates its linear nature. Here:

• \( y \) is the dependent variable.
• \( m \) is the slope, which shows the steepness or incline of the line.
• \( x \) is the independent variable you can change.
• \( b \) is the y-intercept, illustrating where the line crosses the vertical axis.

Linear equations are vital in mathematics for modeling real-life situations where you have a constant rate of change.

Slope-Intercept Form

The slope-intercept form of a linear equation is very useful for graphing and understanding the dynamics of a line. It is written as \( y = mx + b \), where:

• \( m \) denotes the slope, indicating the rate of change or how much \( y \) increases as \( x \) increases.
• \( b \) is the y-intercept, providing a starting point on the y-axis.

Let's break it down further. The slope \( m \) is essentially the change in the y-value for a unit change in x. It answers the question, "How steep is the line?"
Using this form, you can easily graph a line by starting at \( b \) on the y-axis and following the direction given by the slope.
This method is accessible and effective, making it a favorite for quick graphing.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It consists of two number lines intersecting at a right angle:

• The horizontal axis, or x-axis.
• The vertical axis, or y-axis.

These axes divide the plane into four quadrants, each of which can be used to locate points through ordered pairs \((x, y)\).
The origin, where the axes intersect, has the coordinates \((0, 0)\). Any point on this plane is defined by just two numbers, allowing for clear and simple graphing of equations. Moreover, the coordinate plane provides a visual way to understand and analyze relationships and changes represented by equations, such as graphing linear equations like \( f(x) = x - 5 \).

Y-Intercept

The y-intercept is a critical concept in understanding linear equations. It represents the y-coordinate of the point where the line crosses the y-axis. In the equation \( y = mx + b \), the y-intercept is denoted by \( b \).

• This point occurs when \( x = 0 \).
• It helps in quickly sketching the graph of a linear function as it provides an initial point of reference.

For example, in \( f(x) = x - 5 \), the y-intercept is \(-5\). This means the line crosses the y-axis at the point \((0, -5)\).
This straightforward approach to identifying a starting point for graphing is vital. It makes the job of plotting the linear function not only simpler but also more intuitive. Knowing how to find and use the y-intercept allows you to graph effectively and understand the function's behavior on the graph entirely.