Problem 11 - Algebra 1: Concepts and Skills

Question

Graph the equation. \(y=5 x\)

Step-by-Step Solution

Answer

First plot the y-intercept at (0,0). Then from the y-intercept, move up 5 units and to the right 1 unit, plotting a point at this location on the graph. The graph of the equation \(y=5x\) will be a straight line going through these points.

Step 1: Identify the slope and y-intercept

The slope (m) and y-intercept (b) of the linear equation in slope-intercept form \(y=mx+b\) are readily apparent. Here, slope \(m=5\) and \(y\)-intercept \(b=0\). The y-intercept indicates that when \(x=0\), \(y=0\). The slope is the rise over the run, meaning for each unit increase in \(x\), \(y\) increases by 5 units.

Step 2: Plot the y-intercept

Plot the y-intercept (0,0) on the y-axis. This will be the starting point for drawing the line.

Step 3: Plot another point using the slope

From the y-intercept, move up 5 units (due to positive slope) and to the right 1 unit (as the rise/run = 5/1), and plot a point at this location on the graph.

Step 4: Draw the line

Draw a straight line through the points marked. This line represents the equation \(y=5x\).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is crucial for graphing lines efficiently. The general formula for this form is given by \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) represents the y-intercept. This tells us how steep the line is and where it crosses the y-axis. For example, in the equation \(y = 5x\), the slope is 5, which means the line rises 5 units for every unit it runs horizontally. The y-intercept is 0, indicating that the line crosses the y-axis at the origin (0,0). The beauty of the slope-intercept form is its directness; it reveals the line's characteristics without needing extra computation.

When comparing different lines, their slope can tell us a lot. A larger slope means a steeper line, while a slope of zero indicates a horizontal line. If two lines have the same slope, they are parallel. On the other hand, perpendicular lines have slopes that are negative reciprocals of each other.

Plotting Points

Plotting points is the most fundamental skill in graphing equations. Before you can draw a line, you need to determine at least two points that lie on it. Each point is determined by an ordered pair \((x, y)\). When given a linear equation, like our example, \(y = 5x\), we start by plotting the y-intercept, which is our anchor point. Here, the y-intercept is (0,0).

After you have the first point, use the slope to find the next. In simple terms, move vertically by the slope's numerator and horizontally by its denominator. Repeat the process to find more points if needed. A helpful tool for students is to use graph paper, ensuring precise placement of points and ease in creating the slope's rise over run.

Slope and Y-Intercept

The slope and y-intercept are the dynamic duo of the slope-intercept form, offering a snapshot of the line's behavior. The slope, a measure of steepness, shows how much the line ascends or descends for each step to the right. In our exercise, the slope is 5, which can be interpreted as 'go up 5, go right 1'. Slopes can be positive, negative, zero, or undefined. Positive slopes rise to the right, while negative slopes fall to the right. A zero slope means a horizontal line, and an undefined slope signifies a vertical line.

The y-intercept is the point where the line crosses the y-axis. It's always where \(x=0\). Knowing the y-intercept gives us a starting point for drawing the line on the graph. It sets the stage for the story the line is about to tell, how it starts and where it's headed. For the line in the exercise, the y-intercept is at the origin, which is where we begin plotting.

Drawing Straight Lines

After identifying points through the slope and y-intercept, drawing straight lines is the final step in graphing a linear equation. The goal is to connect the dots, or rather, the points you've plotted, with a ruler or a straightedge to create a line that extends in both directions indefinitely.

When teaching, emphasize the need for precision. A common mistake among students is using freehand to draw the line, which can lead to inaccuracies. Remind them that a line represents all the infinite possible solutions to the given equation, and thus, any point on the line should satisfy the equation. Plotting more than two points before drawing the line is a good practice for beginners, to ensure the line's correctness. The line is the culmination of all the parts working together: the slope, the y-intercept, and the points in between.