Problem 5 - Basic Technical Mathematics with Calculus

Question

Display the graphs of the given functions on a graphing calculator. Use appropriate window settings. $$y=3 x-1$$

Step-by-Step Solution

Answer

Graph the linear function \( y = 3x - 1 \) using a window from \( x = -10 \) to 10 and \( y = -10 \) to 10, showing a straight line with slope 3 and y-intercept -1.

Step 1: Identify the Function Type

The given function is a linear equation written in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. For this function, \( y = 3x - 1 \), the slope \( m \) is 3, and the y-intercept \( b \) is -1.

Step 2: Determine Plotting Points

Select easy points to plot on the graph. Start with x = 0: \( y = 3(0) - 1 = -1 \). Thus, the point (0, -1) is on the line. For x = 1: \( y = 3(1) - 1 = 2 \). So, the point (1, 2) is also on the line.

Step 3: Set the Graphing Calculator Window

On a graphing calculator, set the window settings to capture the essential features of the line. Set \( x \)-min and \( x \)-max around -10 and 10, and \( y \)-min and \( y \)-max around -10 and 10. This ensures that the intercept and plotted points are visible.

Step 4: Input the Equation

Enter the equation \( y = 3x - 1 \) into the graphing calculator. This equation falls under the 'Y=' menu where you can directly input the given linear function.

Step 5: Graph the Function

Press the 'Graph' button on the calculator to display the function. Ensure the plotted points (0, -1) and (1, 2) are on the line displayed in the graph screen. Adjust the window settings if necessary to make the line appear well-centered and clearly visible.

Key Concepts

Linear Functions

Linear functions are essential components of algebra, representing straight lines in the Cartesian plane. These functions are expressed in the form \( y = mx + b \), which describes how each value of \( x \) is transformed linearly to produce a \( y \) value. Here, \( m \) is the slope, dictating the steepness of the line, and \( b \) is the y-intercept, showing where the line crosses the y-axis. Understanding linear functions means grasping these fundamental attributes:

• **Slope**: Determines the angle of the line relative to the x-axis. A positive slope means the line ascends, while a negative one descends as you move from left to right.
• **Y-intercept**: The point where the line crosses the y-axis, which occurs when \( x = 0 \).

Linear functions are straightforward, but they are building blocks for more complex mathematical concepts. They model real-world relationships where one quantity depends directly on another, such as speed over time or cost over units. Spend time familiarizing yourself with these basics, as they frequently appear in math problems and everyday situations.

Graphing Calculators

Using graphing calculators transforms the complex task of plotting equations into a simpler procedure. Graphing calculators allow you to enter equations in real-time and see the graphical representation immediately. This visual representation helps solidify the understanding of algebraic functions. When graphing linear equations:

• **Set the window:** Adjust the x-min, x-max, y-min, and y-max values to include critical points on the graph. For instance, a typical range from -10 to 10 is often suitable for linear functions.
• **Input the equation:** Enter your linear equation into the calculator. Most calculators have a 'Y=' menu where this can be done.
• **Graph:** Use the graph button after entering the equation to see it displayed. Adjust the window settings if your line doesn't appear clearly.

Graphing calculators make it easier to visualize trends and better understand equations' behavior, especially when manipulating different linear functions.

Slope-Intercept Form

The slope-intercept form, \( y = mx + b \), is a convenient way to express linear equations. This format is particularly useful because it provides direct information about the line's slope and y-intercept, aiding in quick graphing and analysis.

Understanding Components

• **Slope \( m \):** This dictates how steep the line is. A higher value means a steeper incline, while zero slope indicates a horizontal line.
• **Y-intercept \( b \):** This is where the line will cross the y-axis. Its value directly gives the starting point of your line when \( x = 0 \).

Benefits of Slope-Intercept Form

• **Ease of Graphing:** With values for \( m \) and \( b \), you can quickly plot the y-intercept and use the slope to find another point.
• **Simplifying Calculations:** The formula allows easy adjustment and manipulation of the line, which is particularly helpful in problem-solving.
• **Versatility in Different Contexts:** Suitable for analyzing and comparing various linear relationships quickly.

Mastering the slope-intercept form is key to understanding and graphing linear equations efficiently.