Problem 81 - A Graphical Approach to Precalculus with Limits

Question

Sketch by hand the graph of the line passing through the given point and having the given slope. Label two points on the line. $$\text { Through }(0,-4), m=\frac{3}{4}$$

Step-by-Step Solution

Answer

Sketch a line through (0,-4) with slope 3/4; label points (0,-4) and (4,-1).

Step 1: Understand the Problem

We are tasked with sketching a line that passes through the point \((0, -4)\) with a slope \(m = \frac{3}{4}\). We will need to find a second point on the line to be able to sketch it accurately.

Step 2: Identify the Given Point

The line must pass through the point \((0, -4)\). In the coordinate system, this point represents where the line crosses the y-axis.

Step 3: Use the Slope to Find Another Point

The slope \(m = \frac{3}{4}\) indicates a rise of 3 units for every 4 units of run. Starting at \((0, -4)\), move 4 units to the right (along the x-axis) to \((4, -4)\), and then move 3 units up (along the y-axis) to \((4, -1)\). Thus, \((4, -1)\) is another point on the line.

Step 4: Sketch the Line

Plot both points, \((0, -4)\) and \((4, -1)\), on a coordinate plane. Draw a straight line through these points, extending in both directions.

Step 5: Label the Points

On your sketch, label the points \((0, -4)\) and \((4, -1)\) clearly to indicate they lie on the line.

Key Concepts

Understanding the Slope

The concept of slope is fundamental when working with linear equations. The slope, often represented as \(m\), tells us how steep a line is. It is calculated as the "rise" (the change in the vertical direction) over the "run" (the change in the horizontal direction). For example, a slope of \(\frac{3}{4}\) means that for every 4 units you move horizontally along the x-axis, you move 3 units vertically along the y-axis.
This idea of rise over run allows us to easily determine the direction and steepness of a line. If the slope is positive, as it is in our example, the line will rise as you move from left to right. Conversely, a negative slope would mean the line falls as you move from left to right.
Having this slope allows you to find more points on the line once you know one point. This is crucial for accurately graphing the line on a coordinate plane.

The Coordinate Plane

A coordinate plane is like a map that helps us locate points using pairs of numbers, often noted as \( (x, y) \). Imagine this plane as a sheet of graph paper with x and y axes intersecting at the point \(0, 0\), known as the origin. The x-axis runs horizontally, left and right, while the y-axis runs vertically, up and down.
Each point on this plane is determined by an x-coordinate and a y-coordinate. For instance, the point \((0, -4)\) is located at 0 on the x-axis and -4 on the y-axis. This point is specifically where the line crosses the y-axis.
By plotting points on the coordinate plane, we can visualize the line. With at least two points, such as the ones obtained using the slope, you can sketch a straight line. This visual representation is beneficial for understanding the relationship between the mathematical expression of the line and its graphical representation.

Point-Slope Form of a Line

The point-slope form is one of the common ways to express a linear equation and is particularly useful when you know a point on the line and the slope. The formula is given by \[ y - y_1 = m(x - x_1) \]where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.
In our example, using the point \((0, -4)\) and the slope \(\frac{3}{4}\), you can substitute these values into the formula:\[ y + 4 = \frac{3}{4}(x - 0) \]Simplifying, this becomes:\[ y = \frac{3}{4}x - 4 \].
This equation now describes the line that passes through our given point with the specified slope. By converting the point-slope form to slope-intercept form, \(y = mx + b\), it becomes easier to plot the line on a coordinate plane and understand how it behaves.