Problem 76 - A Graphical Approach to College Algebra

Question

Sketch by hand the graph of the line passing through the given point and having the given slope. Label}\\\ &\text { two points on the line.} \end{aligned} \text { Through }(-2,8), m=-1

Step-by-Step Solution

Answer

Plot (-2,8) and (0,6), then draw the line through these points.

Step 1: Understand the Line Equation

The equation of a line in point-slope form 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. Here, \((x_1, y_1) = (-2, 8)\) and \(m = -1\).

Step 2: Substitute Values

Substitute \((-2, 8)\) into the point-slope form: \( y - 8 = -1(x + 2) \). This gives us an equation that we can use to find other points on the line.

Step 3: Simplify the Equation

Simplify the equation from Step 2 to find the slope-intercept form: \( y = -x + 6 \). This form is useful because it shows the slope \(-1\) and the y-intercept \(6\).

Step 4: Find Another Point

Use the line equation \( y = -x + 6 \) to find another point. Substitute \( x = 0 \) to get \( y = 6 \). This gives another point: \((0, 6)\).

Step 5: Sketch the Line

On graph paper, plot the given point \((-2, 8)\) and the point found in Step 4 \((0, 6)\). Draw a straight line through these points. Label these points on the graph.

Key Concepts

Point-Slope Form

When dealing with linear equations, the **point-slope form** is a very versatile and useful formula. It helps in defining the equation of a line when you have one point on the line and its slope. The general formula is \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) represents a point that the line passes through and \(m\) symbolizes the slope of the line. This format is highly beneficial when you wish to find the equation of a line quickly.
To apply it, simply substitute the coordinates of the given point and the slope into the formula. For example, if the point is \((-2, 8)\) and the slope \(m = -1\), by substituting these values, you get \(y - 8 = -1(x + 2)\). With this setup, you're ready to transition the equation into other forms, find additional points, or graph the line.
Using the point-slope form is like having a "map" for your line. It's precise and delivers exactly what you need to know based on minimal information.

Slope-Intercept Form

The **slope-intercept form** is beloved by students and mathematicians alike for its simplicity and clarity. This form is expressed as \( y = mx + b \), where \(m\) represents the slope and \(b\) the y-intercept. The y-intercept is the point where the line crosses the y-axis, often considered the easiest point to find on a graph.
Converting from point-slope to slope-intercept form is straightforward. From our example, you simplify \( y - 8 = -1(x + 2) \) to \( y = -x + 6 \). Here, the slope is \(-1\) and the y-intercept is \(6\).
This form is particularly convenient for quickly identifying how the line moves and where it sits. It provides an immediate view of both the direction and starting point on the y-axis. For graphing, having the slope and y-intercept readily available speeds up the process of sketching the line.

Graphing Lines

Graphing lines is an essential skill that visualizes linear equations. It turns mathematical expressions into a visual path that can be followed on a graph. To graph a line, you need to identify at least two points and connect them with a straight line.
Using slope-intercept form like \( y = -x + 6 \) simplifies this task. First, you can graph the y-intercept \((0, 6)\). Then, use the slope to find another point. Since the slope is \(-1\), this means you move down 1 unit and right 1 unit from the intercept. This lands you on \((1, 5)\) creating a second point.
- Plot both points on graph paper
- Draw a straight line through these points
- Label these clearly to ensure accuracy
This process not only confirms the solution but also helps visualize how the line changes across the graph, emphasizing the slope's role in the line's direction and steepness. If you ever feel uncertain, graphing is a great way to check your equation's correctness visually.