Problem 27 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(27-38,\) graph each linear equation using the slope and y-intercept $$y=2 x+4$$

Step-by-Step Solution

Answer

The graph of the given equation \(y=2 x+4\) is a straight line with a slope of 2 and passes through the \(y-\)axis at the point (0, 4). The line crosses the y-axis at the point (0, 4) and moves up 2 units for each unit to the right.

Step 1: Identify the Slope and Y-intercept from the given equation

From the given equation, \(y=2 x+4\), the number in front of \(x\) is the slope of the line, and the constant term is the y-intercept. So here, the slope \(m = 2\) and the y-intercept \(c = 4\).

Step 2: Plot the Y-intercept

First, place a point at the y-intercept, which is (0, 4) on the y-axis. This is the point where the line crosses the y-axis.

Step 3: Use the Slope to Find a Second Point

Starting from the y intercept, use the slope to find another point on the line. Since the slope is 2, it means you move 2 units up for each 1 unit you move to the right. You can start from the first point we plotted (0, 4) and move 1 place to the right (this is \(x\)), and 2 places up (this is \(y\)), that gives us the new point (1, 6).

Step 4: Draw the Line

Finally, draw a straight line through these two points. Extend this line so that it crosses the x and y axes.

Key Concepts

Slope and Y-Intercept

Understanding the slope and y-intercept is essential when graphing linear equations. The slope, often denoted as 'm', represents the rate at which the line rises or falls as you move along the x-axis. For example, a slope of 2, like in the equation \(y = 2x + 4\), implies that for every increase of 1 unit along the x-axis, the y-value increases by 2 units, indicating an upward sloping line.

On the other hand, the y-intercept, typically noted as 'b' or 'c' in the equation form \(y = mx + b\), is the point where the line crosses the y-axis. It is the value of 'y' when 'x' is 0. Here, our equation has a y-intercept of 4, meaning the line crosses the y-axis at \((0, 4)\). This starting point is crucial as it is the first point we plot on the graph—providing a reference point for drawing the entire line.

Plotting Points

Plotting points on a graph is a fundamental skill for visualizing linear equations. Start by marking the y-intercept, which is a definitive point you're given. In our equation, we begin with the point \((0, 4)\). Next, use the slope to determine the direction and steepness of the line.

With a slope of 2, we rise up 2 units on the y-axis for each 1 unit we go right on the x-axis. From \((0, 4)\), moving 1 unit right to the point \((1, 4)\), and then up 2 units, lands us at the point \((1, 6)\). These two points, when plotted on a Cartesian plane, should be connected with a straight edge to ensure accuracy. By extending this line, it will show the entire solution set of the equation, following the same pattern no matter how far you extend it.

Linear Equation

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations form straight lines when graphed, and their general form is \(y = mx + b\), where 'm' is the slope and 'b' is the y-intercept.

Such equations represent a direct relationship between 'x' and 'y' where every value of 'x' corresponds to one and only one value of 'y'. This consistency allows us to predict y-values for any x we choose. For the given exercise \(y=2x+4\), the graph will show a clear line illustrating this predictable pattern. Whether equations represent simple situations or complex real-world data, graphing them helps to visualize and understand the relationship between the variables involved.