Problem 78 - 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. $$\text { Through }(-2,-3), m=-\frac{3}{4}$$

Step-by-Step Solution

Answer

Plot points (-2, -3) and (0, -4.5); draw a line through them.

Step 1: Understand the Problem

We need to draw the graph of a line with a given slope and passing through a specific point. The given point is \((-2, -3)\) and the slope \(m\) is \(-\frac{3}{4}\).

Step 2: Use the Point-Slope Form of a Line

The point-slope form equation is \(y - y_1 = m(x - x_1)\). Here, \((x_1, y_1)\) is the point and \(m\) is the slope. Plug in \((x_1, y_1) = (-2, -3)\) and \(m = -\frac{3}{4}\).

Step 3: Substitute the Values

Plugging the values into the point-slope form gives: \[ y + 3 = -\frac{3}{4}(x + 2) \]

Step 4: Simplify the Equation

Distribute \(-\frac{3}{4}\) on the right side: \[ y + 3 = -\frac{3}{4}x - \frac{3}{2} \]Subtract 3 from both sides:\[ y = -\frac{3}{4}x - \frac{3}{2} - 3 \]Combine like terms:\[ y = -\frac{3}{4}x - \frac{9}{2} \]

Step 5: Calculate Additional Points

To find another point on the line, choose a value for \(x\). Let \(x = 0\), then \[ y = -\frac{3}{4} imes 0 - \frac{9}{2} = -\frac{9}{2} \]So, one point is \((0, -\frac{9}{2})\).

Step 6: Sketch the Line

Plot the points \((-2, -3)\) and \((0, -\frac{9}{2})\) on the graph. Draw a straight line through these points, extending it across the graph to show the full line.

Step 7: Label the Points

Label the points on your graph as \((-2, -3)\) and \((0, -\frac{9}{2})\), ensuring they are clearly marked on the line.

Key Concepts

Point-Slope Form of a Line

The point-slope form is a way to describe the equation of a line using a known point and the slope of the line. It is incredibly useful when we want to quickly find the equation of a line without having to know the y-intercept. The form is written as:

• \( y - y_1 = m(x - x_1) \)

Here,

• \((x_1, y_1)\) represents a point the line passes through.
• \(m\) is the slope of the line.

For instance, if you have a line passing through the point \((-2, -3)\) with a slope of \(-\frac{3}{4}\), you would substitute \(-2\) for \(x_1\), \(-3\) for \(y_1\), and \(-\frac{3}{4}\) for \(m\). This gives the equation:

• \( y + 3 = -\frac{3}{4}(x + 2) \)

This format helps quickly convert known pieces of data into a functional equation to further explore more points on the line.

Understanding Slope

The slope of a line, denoted as \( m \), indicates the steepness and direction of the line. It is defined as the "rise over run". This concept comes from how much "rise" (or change in the \(y\)-value) there is for a certain "run" (or change in the \(x\)-value) between two points on a line.

• If the slope is positive, the line rises from left to right.
• If the slope is negative, the line falls from left to right.
• A zero slope indicates a horizontal line, remaining constant.
• An undefined slope suggests a vertical line.

For the slope \(-\frac{3}{4}\), it means for every 4 units moving to the right, the line will drop 3 units downward. Visually, this can often help you determine additional points on a line when graphing.

Coordinate Geometry

Coordinate geometry, often called analytical geometry, allows us to represent geometric shapes like lines using algebraic equations. It relies on a pair of perpendicular axes labeled as the x-axis and the y-axis on a graph.A point in this system, such as \((-2, -3)\), consists of two numbers:

• The first number \(-2\) is the x-coordinate and signifies the horizontal position.
• The second number \(-3\) is the y-coordinate and signifies the vertical position.

By using points and lines, coordinate geometry establishes relationships between algebraic equations and geometric shapes.Graphing lines specifically allows us to visualize how different values in equations translate to geometry. Creating graphs of the equations isn't just about plotting points but understanding how the line continues infinitely, showing continuity and consistency over the coordinate plane. Highlighted points on the graphs, such as \((-2, -3)\) and another calculated point, help solidify this understanding by showing actual intersections on the plane.