Problem 64 - Applied Calculus

Question

Use your graphing calculator to graph the following four equations simultaneously on the window [-10,10] by [-10,10]: $$ \begin{array}{l} y_{1}=3 x+4 \\ y_{2}=x+4 \end{array} $$ \(y_{3}=-x+4\) (Use \([(-)]\) to get \(\left.-x .\right)\) $$ y_{4}=-3 x+4 $$ a. What do the lines have in common and how do they differ? b. Write the equation of a line through this \(y\) -intercept with slope \(\frac{1}{2}\). Then check your answer by graphing it with the others.

Step-by-Step Solution

Answer

All lines intersect at y = 4; they differ in slope. New line: y = \(\frac{1}{2}x + 4\).

Step 1: Input Equations

Begin by turning on your graphing calculator, and access the function input menu. Input each of the given equations: \(y_1 = 3x + 4\), \(y_2 = x + 4\), \(y_3 = -x + 4\), and \(y_4 = -3x + 4\).

Step 2: Set Window Limits

Ensure the graphing window is set up correctly. For this problem, set your window to have an \(x\) range from -10 to 10 and a \(y\) range from -10 to 10.

Step 3: Graph the Equations

With the equations input and the window set, press the graph button to view all four lines simultaneously on the coordinate plane.

Step 4: Analyze the Common Feature

Observe that all lines have the same y-intercept at \(y = 4\). This means all lines intersect the y-axis at this point.

Step 5: Analyze the Differences

Notice that the lines differ in slope. The equations have slopes of 3, 1, -1, and -3, respectively, which affect the steepness and direction of the lines.

Step 6: Find New Line Equation

To find a line with the same y-intercept, \(4\), and a slope of \(\frac{1}{2}\), use the point-slope form. This gives the equation \(y = \frac{1}{2}x + 4\).

Step 7: Graph the New Equation

Enter the new equation \(y = \frac{1}{2}x + 4\) on your graphing calculator and add it to the existing graph to verify alignment with the same y-intercept.

Key Concepts

Understanding Linear Equations

Linear equations are mathematical expressions that describe a straight line on a graph. These equations consist of variables, usually an "x" and "y," with constants that determine the line’s slope and y-intercept.
Linear equations generally take the form of \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept.
They are vital in various fields to describe relationships and predict trends since they are straightforward and easy to manipulate.

• In the exercise, we explored linear equations of the form \(y = 3x + 4\), \(y = x + 4\), \(y = -x + 4\), and \(y = -3x + 4\).

These different lines showcase how altering the slope value changes the line's direction and steepness while keeping the y-intercept constant.

Graphing Functions with a Calculator

Graphing functions, especially linear equations, becomes much simpler with a graphing calculator. These devices aid in visualizing complex relationships between variables, offering a more intuitive learning experience.
To graph a function, follow these steps:

• Turn on your graphing calculator and access the function input menu.
• Enter each equation you wish to graph.

In this exercise, the focus was on graphing \(y = 3x + 4\), \(y = x + 4\), \(y = -x + 4\), and \(y = -3x + 4\) within a window set from \(-10,10\) on both axes. Adjusting the window allows you to fit all relevant parts of the graph within your view.
Once the equations are inputted and the window is set correctly, pressing the graph button lets you visualize how these functions interact in the plane.

The Y-Intercept in Linear Equations

The y-intercept is a crucial concept in graphing linear equations. It refers to the point where the line crosses the y-axis. In the equation \(y = mx + b\), the y-intercept is represented by \(b\), showing where \(x = 0\).
In our exercise, all equations shared a y-intercept of 4, meaning every graph crossed the y-axis at (0, 4).
This common feature is evident when separately plotting \(y = 3x + 4\), \(y = x + 4\), \(y = -x + 4\), and \(y = -3x + 4\); each begins its journey on the y-axis at the same point.
Knowing the y-intercept helps in quickly plotting the line and understanding part of its behavior without complex calculations.

Exploring Slope-Intercept Form

The slope-intercept form of a linear equation is a simple, efficient format for expressing linear relationships. It is written as \(y = mx + b\), with \(m\) as the slope and \(b\) as the y-intercept.
This format makes it straightforward to identify how fast the line increases or decreases (slope) and where it crosses the y-axis (y-intercept).
In the exercise, different slopes (3, 1, -1, and -3) determined the direction and steepness of our lines. Each slope affected how crisp or gradual the incline/decline from the y-intercept was.
For example, \(y = 3x + 4\) has a steeper, more pronounced upward slope compared to \(y = x + 4\), while \(y = -3x + 4\) descends more sharply due to its negative slope. This understanding is fundamental when modeling real-world situations.