Problem 112 - Introductory and Intermediate Algebra for College Students 4th

Question

Use a graphing utility to graph each equation in Exercises in a standard viewing rectangle, \([-10,10,1]\) by \([-10,10,1] .\) Then use the \([\text { TRACE }]\) feature to trace along the line and find the coordinates of two points. $$y=2 x-1$$

Step-by-Step Solution

Answer

The coordinates of two points on the line \(y=2x-1\) in the viewing window \([-10,10,1]\) by \([-10,10,1]\) are (-5,-11) and (3,5).

Step 1: Enter the Equation into the Graphing Calculator

First of all, input the linear equation \(y=2x-1\) into the graphing calculator or utility. Be sure to check that the equation is entered correctly before proceeding.

Step 2: Set the Viewing Rectangle

Next, set the viewing rectangle to \([-10,10,1]\) by \([-10,10,1]\). This scale gives an appropriate view of the line on the graph.

Step 3: Use the TRACE Feature

Now, activate the TRACE feature. Move along the line to identify two points. For example, at \(x=-5\), the y-coordinate is \(y=2(-5)-1=-11\), and at \(x=3\), the y-coordinate is \(y=2(3)-1=5\).

Key Concepts

Graphing Calculator

A graphing calculator is a high-functioning handheld calculator that is capable of plotting graphs, solving simultaneous equations, and performing numerous other tasks with variables. It is an essential tool for students and professionals alike who work with advanced mathematics. When it comes to graphing linear equations such as \(y=2x-1\), the graphing calculator simplifies the process significantly.

First, the user needs to input the equation into the calculator. This usually involves accessing a function or graph mode, where equations can be typed in using a keypad that includes variables like 'x' and 'y'. Once the equation is entered, the graphing calculator can plot the line on a coordinate grid. It's important to ensure that all parentheses and multiplication signs are entered correctly to avoid errors in the graph.

Viewing Rectangle

The viewing rectangle is the portion of the coordinate plane that is displayed on the screen of your graphing calculator. It can be adjusted to better visualize different parts of a graph. For our equation \(y=2x-1\), it is suggested to set the viewing rectangle to a range of \([-10,10,1]\) by \([-10,10,1]\).

This range provides a standard view that fits many functions comfortably on the screen, showing the area where \(x\) and \(y\) values range from -10 to 10. The '1' indicates the scale, meaning that each grid line on the graph represents a single unit step on both the x-axis and y-axis. Adjusting the viewing rectangle can be crucial when analyzing the behavior of a graph as it ensures that key features of the line or curve are visible.

TRACE Feature

The TRACE feature on a graphing calculator allows the user to move along the plotted graph and obtain coordinates of specific points. After entering the linear equation and setting the appropriate viewing rectangle, you can activate the TRACE function. This feature is particularly helpful when you need to find precise points on the graph that are not necessarily where the graph intersects the axes or at integer values.

For instance, with the equation \(y=2x-1\), you might want to find specific y-values for chosen x-values that are not easily calculated by hand. By using the TRACE feature, as you move along the graph, the calculator provides the exact coordinates, allowing you to find points such as \((-5,-11)\) or \((3,5)\), offering additional insight into the behavior of the equation on the graph.

Coordinate System

The coordinate system is a method of representing points in a space of given dimensions by coordinates. It is a fundamental concept in graphing as it is used to visualize relationships between variables, such as in a linear equation. The two-dimensional Cartesian coordinate system, composed of an x-axis (horizontal) and a y-axis (vertical), is typically used in graphing linear equations on a graphing calculator.

In this coordinate system, any point on the plane is described by an ordered pair of numbers, also known as coordinates. For example, the point \((3,5)\) indicates that from the origin (0,0), you would move 3 units to the right along the x-axis and 5 units up along the y-axis. Graphing the linear equation \(y=2x-1\) on this system results in a straight line, where every point on the line follows the rule set by the equation.