Problem 56 - Applied Calculus

Question

a. Graph the lines \(y_{1}=x+2, \quad y_{2}=x+1\), \(y_{3}=x, \quad y_{4}=x-1,\) and \(y_{5}=x-2\) on the window [-5,5] by [-5,5] . Observe how the constant changes the position of the line. b. Predict how the lines \(y=x+4\) and \(y=x-4\) would look, and then check your prediction by graphing them.

Step-by-Step Solution

Answer

The constant in each equation shifts the line vertically. Adding a constant moves the line up, subtracting moves it down. Lines \(y = x + 4\) and \(y = x - 4\) confirm this.

Step 1: Understand the Equations

We have five equations: \(y_1 = x + 2\), \(y_2 = x + 1\), \(y_3 = x\), \(y_4 = x - 1\), and \(y_5 = x - 2\). Each equation represents a line with a slope of 1, since the coefficient of \(x\) is 1, but each has a different constant. The constant represents the y-intercept, where the line crosses the y-axis.

Step 2: Plot the Lines

Plot each line on the same coordinate grid with the window [-5,5] by [-5,5]. For \(y_1 = x+2\), the line crosses the y-axis at (0, 2). Each subsequent line \(y_2, y_3, y_4, y_5\) crosses the y-axis one unit lower than the previous, showing how the constant affects vertical shift.

Step 3: Analyze the Changes

As you compare the graphs, observe that increasing the constant in the equation by 1 shifts the line upwards by one unit without changing its slope. Conversely, decreasing the constant by 1 shifts the line downwards by one unit.

Step 4: Make Predictions for New Lines

For \(y = x + 4\), predict the line will cross the y-axis at (0, 4) and will be parallel to the other lines. For \(y = x - 4\), predict it will cross the y-axis at (0, -4) and also be parallel to the other lines.

Step 5: Graph New Lines

Plot the new lines \(y = x + 4\) and \(y = x - 4\) on the same grid. Confirm that \(y = x + 4\) is shifted upwards by 4 units from \(y = x\), and \(y = x - 4\) is shifted downwards by 4 units. Both lines maintain a slope of 1, confirming the pattern observed.

Key Concepts

Y-Intercept

In a linear equation like \( y = mx + b \), the y-intercept is the point where the line crosses the y-axis. This occurs when the value of \( x \) is zero. Therefore, the y-intercept is represented by the constant \( b \) in the equation.
For instance, in the equation \( y = x + 2 \), the y-intercept is 2, which means that the line crosses the y-axis at the point (0, 2).

• Key idea: The y-intercept indicates the starting value of the dependent variable when the independent variable is zero.
• It provides a starting point for graphing the line on the coordinate plane.

Understanding the y-intercept is crucial for predicting how the line will appear on a graph, especially in relation to other lines.

Slope

The slope of a line is a measure of its steepness and direction. In the standard linear equation \( y = mx + b \), the coefficient \( m \) represents the slope. It tells us how much \( y \) changes for a given change in \( x \).
For example, a slope of 1 means that for every one unit increase in \( x \), \( y \) increases by one unit as well. This results in a line that ascends at a 45-degree angle.

• Positive slope: Line rises as you move from left to right.
• Negative slope: Line falls as you move from left to right.
• Zero slope: Creates a horizontal line.

Understanding slope is essential for comparing lines. In our example, all lines have a slope of 1, meaning they are parallel to one another.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can graph equations. It is defined by two perpendicular number lines: the horizontal x-axis and the vertical y-axis.
This plane allows us to visually represent relationships between variables by plotting points, lines, and curves. The intersection of the x-axis and y-axis is the origin, a central point denoted as (0, 0).

• The x-axis represents the independent variable.
• The y-axis represents the dependent variable.
• The plane enables easy reading of coordinates and visualization of geometric relationships.

Using a consistent scale along both axes, such as the window [-5, 5] by [-5, 5], helps to accurately represent lines and observe shifts.

Graphing Lines

Graphing lines on a coordinate plane involves plotting points and drawing a straight path through these points. Steps to graph include identifying the y-intercept, using the slope to find additional points, and then connecting these points.
For the equations \( y = x + 2 \), \( y = x + 1 \), and so on, start each line by marking its y-intercept on the y-axis. Use the slope, which is 1 in these cases, to find another point by moving up one unit and right one unit, forming a path parallel to the other lines.

• Consistent slopes create parallel lines.
• Graphing allows comparison of different functions and the effect of changing constants.
• Proper scaling ensures accurate representation of lines.

By systematically plotting and connecting points, you create a visual representation of the linear equations, demonstrating the impact of their y-intercepts and parallel nature due to identical slopes.