Problem 32 - Algebra and Trigonometry Real Mathematics, Real People

Question

Using a Graphing Utility to Graph an Equation In Exercises \(31-44,\) use a graphing utility to graph the equation. Use a standard viewing window. Approximate any \(x\) - or \(y\) -intercepts of the graph. \(y=\frac{2}{3} x-1\)

Step-by-Step Solution

Answer

The x-intercept of the graph is 1.5 and the y-intercept is -1.

Step 1: Enter the equation into the graphing utility

First, the equation \(y=\frac{2}{3} x-1\) should be entered into the graphing utility. Make sure the equation is correct before proceeding.

Step 2: Graph the equation

After entering the equation, press the graph button to display the graph of this equation. The graph will represent a straight line due to the nature of the equation (it is a linear equation). The line should slant upwards because the coefficient of x is positive.

Step 3: Identify the y-intercept

The y-intercept is where the line crosses the y-axis. In this equation, the y-intercept is -1, as is evident from the equation in the form \(y=mx+c\) where \(c\) is the y-intercept.

Step 4: Identify the x-intercept

The x-intercept is where the line crosses the x-axis. This occurs when \(y=0\). To find the x-intercept, replace \(y\) in the equation with 0, and solve for \(x\). This gives the equation \(0=\frac{2}{3}x -1\) which solves to \(x=1.5\).

Key Concepts

Graphing Utility

Graphing utilities are powerful tools that facilitate the process of visualizing equations for students. They turn complex algebraic expressions into comprehensible visual graphs. To use a graphing utility, one must first input the given equation, such as \(y=\frac{2}{3}x-1\), ensuring that the format aligns with the software's requirements. After entering the equation, the utility typically provides an option to display the graph.

For beginners, it's crucial to get familiar with the interface and functions of the graphing utility. Experimenting with different types of equations helps to understand how changes in the equation affect the shape and the position of the graph.

Y-intercept

Understanding the Y-intercept

The y-intercept is a foundational concept in graphing linear equations. It's defined as the point where the graph of the equation crosses the y-axis. In essence, it represents the value of \(y\) when \(x=0\). In the equation provided, \(y=\frac{2}{3}x-1\), the y-intercept is -1. This means that the line will cross the y-axis at the point \(0, -1\).

Knowing the y-intercept is particularly useful when graphing by hand or for a quick sketch to understand the behavior of the graph.

X-intercept

Identifying the X-intercept

Conversely, the x-intercept is the point at which the graph crosses the x-axis, or where \(y=0\). To determine the x-intercept algebraically, set the \(y\) value in the equation to zero and solve for \(x\). For our example \(y=\frac{2}{3}x-1\), setting \(y\) to zero yields \(x=1.5\), so the line crosses the x-axis at \(1.5, 0\). Recognizing the x-intercept offers insight into the roots of the equation and is essential when sketching the behavior of the line on graph paper or using a graphing utility.

Standard Viewing Window

What is a Standard Viewing Window?

A standard viewing window in graphing utilities is a pre-set screen size that displays the graph. It’s designed to show the most relevant portion of the plane for typical equations. For our purposes, we generally consider a standard window to include values from -10 to 10 on both the x and y axes.

This window size is typically sufficient to visually capture the y- and x-intercepts for basic linear equations like \(y=\frac{2}{3}x-1\). Starting with a standard window helps in maintaining consistency when comparing graphs and ensures that key features of the graph are not missed.

Linear Equation

Basics of a Linear Equation

A linear equation, such as \(y=\frac{2}{3}x-1\), represents a straight line when graphed on a coordinate plane. The equation is in the slope-intercept form \(y=mx+c\), where \(m\) is the slope and \(c\) is the y-intercept. The slope indicates the steepness and direction of the line: a positive slope means the line ascends to the right, while a negative slope means it descends.

Linear equations are crucial in understanding relationships between two variables, providing a clear visual representation of how one variable affects another. For students, grappling with linear equations is essential, as they form the foundation for more complex mathematical concepts.