Problem 52 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the function. \(f(x)=4 x+2\)

Step-by-Step Solution

Answer

The graph of the function \(f(x)=4x+2\) is a straight line that passes through the y-intercept (0,2) and rises by 4 units for every unit it moves to the right.

Step 1: Definition of terms

The given function is \(f(x)=4x+2\). This is a linear equation. Therefore, it is in the form \(mx+b\) where \(m\) is the slope of the line and \(b\) is the y-intercept.

Step 2: Identify the slope and the y-intercept

In the equation \(f(x)=4x+2\), the slope \(m=4\) and the y-intercept \(b=2\). This means that the line rises by 4 units for every unit it moves to the right. In addition, the line crosses the y-axis at the point (0,2).

Step 3: Plotting the graph

To draw the graph, place a point at the y-intercept (0,2) on the y-axis. From there, move 4 units up and 1 unit to the right and mark another point. Repeat until you have enough points to draw a line. The line that passes through all these points represents the function \(f(x)=4x+2\) on the graph.

Key Concepts

Linear Function Representation

Understanding how a linear function is represented on a graph is integral in mathematics. A linear function is one of the simplest forms of a function in algebra. It creates a straight line when plotted on a graph. The general form of a linear equation is given by the formula
\( y = mx + b \)
where
\( y \) represents the dependent variable,
\( x \) represents the independent variable,
\( m \) indicates the slope of the line, and
\( b \) signifies the y-intercept, or the point where the line crosses the y-axis. In the case of \( f(x)=4x+2 \) given in the exercise, the equation is telling us how to take any value of \( x \) and find its corresponding value of \( f(x) \) (or \( y \) in traditional graphing).
Visualizing this on a coordinate plane, every combination of \( x \) and \( f(x) \) creates a point. When all possible points are connected, they form a straight line, perfectly encapsulating the concept of a linear function.

Slope-Intercept Form

The slope-intercept form is incredibly useful for quickly sketching the graph of a linear equation. This form, written as \( y = mx + b \), clearly displays the two most critical features for graphing: the slope and the y-intercept. Within this context,
\( m \) is the slope: it tells us how steep the line is, and whether it ascends or descends as we move from left to right across the graph. A positive slope means the line rises, whereas a negative slope means it falls. The number itself signifies how much the line rises (or falls) for a given horizontal movement.
\( b \) is the y-intercept, marking the exact point where the line will cross the y-axis, occurring when \( x = 0 \). This starting point is crucial as it is the initial reference from which we apply the slope to continue plotting the line. For instance, with a slope of 4, as in the exercise \( f(x)=4x+2 \), it tells us that for each single unit you move to the right along the x-axis, you need to move up 4 units along the y-axis to follow the line's path. Thus, the slope-intercept form not only provides a clear recipe for plotting a linear equation but also greatly simplifies the process of understanding the behavior of the line.

Y-Intercept of a Line

The y-intercept is a fundamental feature on the graph of a linear equation. It is where the line crosses the y-axis, and in the slope-intercept equation \( y = mx + b \), it's represented by \( b \). This particular point has an x-coordinate of 0, indicating that it captures the value of \( y \) when \( x \) is absent. To plot the y-intercept, you simply locate the value of \( b \) up (or down if it's negative) the y-axis.
From there, the y-intercept serves as a starting point for graphing the whole line. In our example \( f(x)=4x+2 \), the y-intercept is 2, meaning that the line will pass through the point \( (0,2) \). After marking this initial point, you can then apply the slope (increment of 4 in y per unit of x) to find additional points. Remember, the y-intercept is only one point on the line, but it's crucial because it anchors the line at a known position from which we can apply the slope to continue the line in either direction across the graph.