Problem 33 - A Graphical Approach to College Algebra

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. 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. $$y=3 x$$

Step-by-Step Solution

Answer

Plot (0, 0) and (1, 3); draw a line through them.

Step 1: Identify Intercept

The given equation is in the form of a line equation: \(y = kx\). Here, \(k=3\), so the equation is \(y = 3x\). Since \(y= kx\) for any real number \(k\) always has the origin (0, 0) as a point on the line, the y-intercept is at the origin (0, 0).

Step 2: Choose a Second Point

To find a second point on the line, substitute a simple value for \(x\) (typically 1 to keep computations easy). Substitute \(x = 1\) into the equation \(y = 3x\): thus, \(y = 3 \times 1 = 3\). This gives us the point (1, 3).

Step 3: Plot Points

Plot the origin (0, 0) and the second point (1, 3) on the graph.

Step 4: Draw the Line

Join the two points (0, 0) and (1, 3) with a straight line to represent the graph of the equation \(y = 3x\).

Key Concepts

Graphing Functions

Graphing functions provides us a visual representation of equations, helping to understand their behavior more intuitively. When we graph a function like a linear equation, we plot points on a coordinate plane. These points are then connected to reveal the function's shape.

For the equation \(y = 3x\), graphing it involves plotting points that satisfy this equation. In this case, the origin \((0, 0)\) and another point obtained by choosing an arbitrary \(x\)-value. Substituting \(x = 1\) gives us the point \((1, 3)\).

These points help structure the line graphically, translating the equation into a tangible form. The simplicity of graphing relies on recognizing that every solution to the equation corresponds to a point on the graph.

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most common ways to represent linear equations: \(y = mx + b\). In this form, \(m\) represents the slope of the line, and \(b\) indicates the \(y\)-intercept. For the equation \(y = 3x\), we can observe that:

• The slope \(m\) is 3, which tells us how steep the line is.
• There is no \(b\) value (it's implicitly zero), so the line passes through the origin.

The slope informs us that for every increase of one unit along the \(x\)-axis, the \(y\) value increases by three units. This consistent change is why the line maintains its straight appearance. Recognizing lines through their slope-intercept form can simplify graphing and predicting behavior of linear functions.

Coordinate Geometry

Coordinate geometry, or analytic geometry, melds algebra and geometry using a coordinate plane. It allows us to describe and analyze geometric figures numerically.

In coordinate geometry, each point on the plane is defined by an \((x, y)\) coordinate. This system makes it easier to plot and understand lines, like \(y = 3x\). By identifying points, such as \((0, 0)\) and \((1, 3)\), we apply algebraic understanding to geometric representation.

With coordinate geometry, essential concepts like distance, slope, and midpoint become calculations on a plane, leading to deeper insights into shapes and their properties. This blend of algebraic equations with geometric visuals guides learners in building a comprehensive understanding of mathematical relationships.