Problem 36 - 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. $$y=1.5 x$$

Step-by-Step Solution

Answer

The line passes through the origin \((0,0)\) and point \((2,3)\), forming a straight line.

Step 1: Identify Given Equation

We are given the equation of the line: \(y = 1.5x\). This equation represents a straight line where \(k = 1.5\) is the slope of the line.

Step 2: Confirm Line Passes through Origin

For the line \(y = 1.5x\), when \(x = 0\), we substitute into the equation to find \(y\). \[ y = 1.5 \times 0 = 0 \]Thus, the line passes through the origin \((0, 0)\).

Step 3: Determine a Second Point

To determine another point on the line, choose any value for \(x\). Let's choose \(x = 2\). Substitute \(x\) into the equation:\[ y = 1.5 \times 2 = 3 \]Therefore, the second point is \((2, 3)\).

Step 4: Plot Points and Draw the Line

On a coordinate plane, plot the origin \((0, 0)\) and the second point \((2, 3)\). Use a ruler to draw a straight line connecting these two points. This line represents the equation \(y = 1.5x\).

Key Concepts

Slope

The slope in the context of a line on a graph is a measure of how steep the line is. It is often represented by the letter \(m\) or, in our original exercise, it is given by the value \(k\). The slope tells us how much the \(y\)-value (or vertical value) changes for every one unit change in the \(x\)-value (or horizontal value).

• If the slope \(k\) is positive, the line will ascend, meaning it goes upwards as it moves from left to right.
• If the slope is negative, the line will descend, moving downwards from left to right.
• If the slope is zero, the line is horizontal and does not rise or fall.

The slope is calculated as \(\frac{\text{rise}}{\text{run}}\), commonly rewritten as \(\frac{\Delta y}{\Delta x}\). Understanding the slope helps predict how the line will behave as \(x\) varies. In the exercise, the line \(y = 1.5x\) has a slope of 1.5, indicating that for every 1 unit the line moves horizontally, it rises vertically by 1.5 units.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can graph equations. It consists of two perpendicular axes: the horizontal axis is the \(x\)-axis, and the vertical axis is the \(y\)-axis. These axes intersect at the coordinate plane's origin, which is the point \((0, 0)\).

• Each point on the plane is identified by a pair of numerical coordinates \((x, y)\), with the \(x\)-coordinate showing the position relative to the \(x\)-axis, and the \(y\)-coordinate showing the position relative to the \(y\)-axis.
• The coordinate plane is divided into four quadrants, each defined by the signs of \(x\) and \(y\).
• The first quadrant has both coordinates positive, while the second has a negative \(x\) and positive \(y\).

Graphing lines on this plane involves plotting points that satisfy the equation, then drawing a line through those points. By using the origin \((0, 0)\) and another determined point such as \((2, 3)\), you can accurately depict the line \(y = 1.5x\) as shown in the original exercise.

Equation of a Line

The equation of a line in mathematics often defines the relationship between \(x\) and \(y\) in a linear form. One common format for the equation of a line is \(y = mx + b\), where \(m\) is the slope, and \(b\) is the \(y\)-intercept, which is the point where the line crosses the \(y\)-axis.

• In equations like \(y = kx\), where there is no \(b\) term, the line passes through the origin \((0, 0)\) because this point satisfies the equation when \(x = 0\).
• The presence of \(k\) in \(y = kx\) determines the steepness of the line, identical to the slope.
• For example, the equation \(y = 1.5x\) implies the line passes through the origin and slopes upwards, with \(y\) increasing by 1.5 for every increase of 1 in \(x\).

Graphs of linear equations can easily be constructed once the slope \(m\) and \(y\)-intercept \(b\) are understood, allowing you to create accurate and informative visual representations of equations.