Problem 32 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the given equation. Label the intercepts. $$y=3 x$$

Step-by-Step Solution

Answer

The graph of y = 3x is a line passing through the origin (0, 0) with a slope of 3.

Step 1: Understand the Equation

The given equation is a linear equation in the form of y = mx + c, where m is the slope, and c is the y-intercept. Here, y = 3x, so the slope m = 3 and the y-intercept c = 0.

Step 2: Determine the Intercepts

To find the intercepts, first find the y-intercept: Setting x = 0 in the equation y = 3x gives y = 0. Therefore, the y-intercept is (0, 0). To find the x-intercept, set y = 0, which also gives x = 0. Thus, the x-intercept is (0, 0).

Step 3: Plot the Intercepts

On a coordinate plane, plot the intercepts found in the previous step. The point (0, 0) should be marked.

Step 4: Plot Additional Points

To accurately sketch the line, select another value for x. For example, if x = 1, then y = 3(1) = 3, so the point (1, 3) can also be plotted. Similarly, if x = -1, then y = 3(-1) = -3, plot the point (-1, -3).

Step 5: Draw the Line

Connect the points (0, 0), (1, 3), and (-1, -3) with a straight line. This line represents the graph of the equation y = 3x.

Step 6: Label the Intercepts

Clearly label the intercept at (0, 0) on the graph to show where the line crosses the axes.

Key Concepts

Coordinate Plane

A coordinate plane is like a map. It helps you locate points using two numbers: the x-coordinate and the y-coordinate. The plane is divided into four quadrants by a horizontal line called the x-axis and a vertical line called the y-axis. The point where these two axes meet is called the origin, represented as (0, 0).

To find any point on this plane, you look at how far it is from the origin in the horizontal direction (x-coordinate) and the vertical direction (y-coordinate). For example, the point (2, 3) is 2 units to the right of the origin and 3 units up.

Intercepts

Intercepts are where a line crosses the axes on a coordinate plane.

The x-intercept is where the line crosses the x-axis, meaning the y value is zero at this point. Conversely, the y-intercept is where the line crosses the y-axis, so the x value is zero here.

In the given equation y = 3x, both the x-intercept and y-intercept are at (0, 0). This means the line goes through the origin.
To find the intercepts:

• Set x = 0 to find the y-intercept.
• Set y = 0 to find the x-intercept.

Slope

The slope of a line explains its steepness and direction.

It's represented by 'm' in the linear equation y = mx + c. A positive slope means the line goes up from left to right, and a negative slope means it goes down.

The slope is typically calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In simpler terms, it's: m = (change in y) / (change in x).
In our equation y = 3x, the slope is 3. This means for every unit you move to the right on the x-axis, you'll move up 3 units on the y-axis.

Linear Equation

A linear equation forms a straight line when graphed on a coordinate plane.

It's usually written in the form y = mx + c, where:

• y is the dependent variable (output)
• x is the independent variable (input)
• m is the slope
• c is the y-intercept

Linear equations can have one variable or two, and they show a constant rate of change. In the equation y = 3x, the y-intercept is 0 and the slope is 3, simplifying it to y = 3x, indicating a line passing through the origin with a steepness of 3 units up for every unit right.