Problem 35 - A Graphical Approach to Precalculus with Limits

Question

A line having an equation of the form \(y=k x\), where \(k\) is a real number, \(k \neq 0\), will always pass through the origin \((0,0) .\) To graph such an equation by hand, we can determine a second point and then join the origin and that second point with a straight line. Use this method to graph each line. $$y=-0.75 x$$

Step-by-Step Solution

Answer

Plot the points (0,0) and (4,-3), then draw the line through them.

Step 1: Identify the Slope

The equation of the line is given as \( y = -0.75x \). In this equation, the slope \( k \) is -0.75, indicating that for every unit increase in \( x \), \( y \) decreases by 0.75 units due to the negative sign.

Step 2: Identify the Point at the Origin

Since the equation is in the form \( y = kx \), the line always passes through the origin point \((0,0)\). This will be one of the points you use to graph the line.

Step 3: Calculate a Second Point

To find another point on the line, choose a simple value for \( x \). Here, let's take \( x = 4 \). Substitute \( x = 4 \) in the equation: \( y = -0.75(4) = -3 \). Thus, the second point on the line is \((4, -3)\).

Step 4: Plot the Points on a Graph

Draw a graph and mark the points \((0,0)\) and \((4,-3)\) on it. These two points are sufficient to draw the line because a line is determined by two points.

Step 5: Draw the Line

Connect the two points—\((0,0)\) and \((4,-3)\)—with a straight edge, creating the line representing the equation \( y = -0.75x \). The line should extend beyond these points in both directions to properly show it along the graph's grid.

Key Concepts

Slope and Y-intercept

The slope and y-intercept are fundamental properties when dealing with linear equations of the form \( y = kx + b \). In this type of equation, the slope \( k \) is a number that tells us how steep the line is.

• A positive slope means the line rises from left to right.
• A negative slope, like in our example \( y = -0.75x \), means the line falls from left to right.

The y-intercept \( b \) is where the line crosses the y-axis. However, in our exercise, the equation \( y = -0.75x \) actually has \( b = 0 \), meaning it crosses through the origin \((0,0)\). This is because it simplifies to the form \( y = kx \), which directly indicates that the point at the origin is part of the line. You can always determine the y-intercept from the equation by setting \( x = 0 \).
The slope \( k \) in \( y = -0.75x \) indicates each increase by 1 unit in \( x \) corresponds to a decrease of 0.75 units in \( y \) due to the negative sign. Understanding the role of the slope helps in predicting the behavior of the line on the coordinate plane.

Coordinate Plane

The coordinate plane is like a map where you can find locations using two numbers: the x-coordinate and the y-coordinate. These coordinates tell you how far to go from the origin point (0,0), which is where the x-axis and y-axis intersect, and both values are zero.
Think of the x-axis as the road going left and right, and the y-axis as the road going up and down.

• The x-coordinate tells you how many steps to move left or right from the origin.
• The y-coordinate tells you how many steps to move up or down from the origin.

In our example of graphing the line \( y = -0.75x \), the line passes through the origin \((0,0)\) since both x and y start at zero. The coordinate plane gives a visual representation to understand where each point is placed and how the line forms by connecting these points.
Lines that move across this plane will adjust depending on their slope \( k \), and their direction will change based on whether they interact with any point other than the origin.

Graphing Techniques

Graphing techniques are essential to accurately draw linear equations and understand their behavior. The steps are straightforward but crucial for accurate visualization. First, plot the origin (0,0) on the graph. This is your starting point since it's the specific feature of the equation in the form \( y = kx \). Next, to establish another point, we choose a specific value for \( x \).
For our equation \( y = -0.75x \), we picked \( x = 4 \). Plugging \( x = 4 \) into the equation gives us a corresponding \( y \)-value of \(-3\), resulting in the point \( (4, -3) \). Now, plot \( (4, -3) \) on the graph.

• Use a ruler for accuracy, and draw a straight line through both points.
• Remember, a line extends infinitely, so let it stretch beyond your two points.

This ensures the line represents all the potential solutions to the equation. Practicing these graphing techniques gives students confidence in plotting any linear equation they encounter.