Problem 35 - A Graphical Approach to College Algebra

Question

Give two parametric representations for each plane curve. Use your calculator to verify your results. $$y=2 x+3$$

Step-by-Step Solution

Answer

Parameters: \((x, y) = (t, 2t + 3)\) and \((x, y) = \left( \frac{t - 3}{2}, t \right)\).

Step 1: Understand the Equation of the Line

The given equation \( y = 2x + 3 \) represents a straight line in the Cartesian coordinate system. The slope of the line is the coefficient of \( x \), which is 2, and the y-intercept is 3.

Step 2: Parametric Representation with x as Parameter

In parametric representation, we express both \( x \) and \( y \) in terms of a parameter, often \( t \). One straightforward way is to let \( x = t \). Then, substituting into the line equation, we have \( y = 2t + 3 \). Therefore, one parametric representation is \( (x, y) = (t, 2t + 3) \).

Step 3: Parametric Representation with y as Parameter

Alternatively, we can use \( y \) as the parameter. Let \( y = t \). In this case, solve for \( x \) from the given line equation: \( t = 2x + 3 \). Rearrange to get \( x = \frac{t - 3}{2} \). Thus, the second parametric representation is \( (x, y) = \left( \frac{t - 3}{2}, t \right) \).

Step 4: Verify Using a Calculator

To verify, plug values of \( t \) into both parametric equations to get coordinates that should satisfy the original equation \( y = 2x + 3 \). Check several values, confirming the outputs adhere to the line equation.

Key Concepts

Cartesian Coordinate System

In mathematics, the Cartesian coordinate system is a fundamental system for describing geometric shapes. Named after René Descartes, it allows for the representation of geometric figures using algebraic equations. This system uses two perpendicular axes, usually labeled as the x-axis and y-axis, which intersect at a point called the origin. The origin is the point (0, 0).

Each point in the plane is given by an ordered pair of numbers \(x, y\), where \x\ represents the horizontal distance from the origin and \y\ the vertical distance from the origin. This system can easily convert geometric problems into algebraic expressions, making calculations more straightforward.

• The x-coordinate indicates the point's position along the horizontal axis.
• The y-coordinate indicates the point's position along the vertical axis.
• These coordinates can describe any point uniquely in a two-dimensional space.

Equation of a Line

An equation of a line provides a way to mathematically describe a straight line within the Cartesian coordinate system. One of the most common forms of the equation of a line is the slope-intercept form, written as \(y = mx + b\). Here, \m\ represents the slope of the line, and \b\ is the y-intercept.

This linear equation helps to determine the relationship between the x and y coordinates on a line.

• It shows how the y value changes as the x value shifts.
• The given equation \(y = 2x + 3\) is a classic example of a linear equation in slope-intercept form.

Understanding this equation allows you to visualize the line it represents and predict y values for any given x.

Slope

The slope of a line is a measure of its steepness. It is usually denoted by the letter \m\ in the line equation. In the formula \(y = mx + b\), the slope is \m\. It tells you how much y increases for a unit increase in x.

The slope can be seen as the "rise over run," which is the vertical change divided by the horizontal change between two points on the line.

• In the example equation \(y = 2x + 3\), the slope is 2.
• This indicates that for every 1 unit increase in x, y increases by 2 units.

The slope's value also indicates the direction of the line:

• A positive slope means the line ascends from left to right.
• A negative slope means it descends.
• A zero slope represents a horizontal line.
• An undefined slope corresponds to a vertical line.

y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept form of a line, \y = mx + b\, the y-intercept is represented by \b\. This point is crucial as it provides a starting point for graphing the line.

• In the equation \(y = 2x + 3\), the y-intercept is 3.
• This means that at the point (0, 3), the line intersects the y-axis.
• The y-intercept is where x equals zero and gives the value of y at that point.

Understanding the y-intercept helps to quickly determine one of the vital characteristics of the line. It simplifies sketching and analyzing the graph by providing a fixed reference from which to draw the line based on the slope.