Problem 43 - College Algebra Essentials

Question

Give the slope and \(y\) -intercept of each line whose equation is given. Then graph the linear function. $$f(x)=\frac{3}{4} x-2$$

Step-by-Step Solution

Answer

The slope of the line is \frac{3}{4} and the y-intercept of the line is -2.

Step 1: Identify the Slope

The slope of the line, denoted as \(m\), is the coefficient of \(x\) in the equation. In the function \(f(x)=\frac{3}{4} x-2\), the slope \(m\) is \frac{3}{4}.

Step 2: Identify the y-intercept

The y-intercept of the line, denoted as \(c\), is the constant in the equation. In the function \(f(x)=\frac{3}{4} x-2\), the y-intercept \(c\) is -2.

Step 3: Graph the Function

Plot the y-intercept point \(-2\) on the y-axis. Since the slope is \frac{3}{4}, for every 4 units move to the right on the x-axis (increasing x), move 3 units upwards (increasing y) from y-intercept point(-2). This will give another point on the graph. Draw a straight line through these two points. This line represents the function \(f(x)=\frac{3}{4} x-2\).

Key Concepts

Slope of a Line

The slope is the measure of the steepness of a line, commonly denoted as 'm'. It is calculated as the ratio of the rise (vertical change) to the run (horizontal change) between two points on a line. In simpler terms, slope tells you how quickly y increases or decreases as x increases.

Considering the equation of the function given by \(f(x)=\frac{3}{4} x-2\), the slope is the coefficient of \(x\) in the equation, which in this case is \(\frac{3}{4}\). This means for every increase of 4 units in \(x\), the \(y\) value will increase by 3 units. If the slope is positive, as it is here, the line ascends from left to right; if it were negative, the line would descend.

A good way to envision this is by visualizing climbing a hill—the steeper the hill, the greater the slope. In practical terms, if you were to walk along this line on a graph, starting from the y-intercept, you would move four steps to the right and three steps up repeatedly to follow the line's path.

Y-Intercept

The y-intercept of a line refers to the point where the line crosses the y-axis; this is the value of \(y\) when \(x = 0\). It is a useful starting point for graphing linear equations and is generally represented as the 'b' (or sometimes 'c') in the standard line equation \(y = mx + b\).

In the example \(f(x)=\frac{3}{4} x-2\), the constant term '-2' represents the y-intercept. This indicates that the line will cross the y-axis at the point \((0, -2)\). When you're starting to graph a linear function, you first plot this point on the y-axis. Afterward, use the slope to find other points by moving from this initial intercept point according to the slope's rise over run ratio.

Linear Equations

A linear equation represents a straight line when plotted on a graph and can be written in various forms, with the most common being the slope-intercept form \(y = mx + b\), where 'm' is the slope and 'b' is the y-intercept.

The given equation, \(f(x)=\frac{3}{4} x-2\), is in this slope-intercept form. Here, 'x' and 'y' or in this case \(f(x)\), are variables that denote any point on the line. Linear equations are fundamental in algebra because they're used to model real-world situations with a constant rate of change. It's also worth noting that their graph will always result in a straight line—there will never be curves or bends.

To effectively understand and manipulate these equations, remember that changing the slope affects the angle of the line while altering the y-intercept shifts the entire line vertically up or down without changing its angle. Being able to identify these two components is critical to mastering the basics of linear functions.