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Chapter 2: Problem 31

Use a graphing calculator to graph each equation in the standard viewing window. $$3 x+4 y=6$$

Short Answer

Expert verified
Rewrite equation as \(y = -\frac{3}{4}x + \frac{3}{2}\), graph it using a graphing calculator.

Step by step solution

01

- Rewrite the equation in slope-intercept form

Start by rewriting the given equation in the form of \(y = mx + b\). The given equation is \(3x + 4y = 6\). Solve for \(y\) by isolating it on one side of the equation.
02

- Solve for y

Subtract \(3x\) from both sides: \(4y = -3x + 6\). Then divide all terms by 4: \(y = -\frac{3}{4}x + \frac{6}{4}\) or \(y = -\frac{3}{4}x + \frac{3}{2}\).
03

- Enter the equation into the graphing calculator

Input the equation \(y = -\frac{3}{4}x + \frac{3}{2}\) into the graphing calculator. Make sure the calculator is set to the standard viewing window, which typically ranges from \(-10, 10\) on both the x-axis and y-axis.
04

- Graph the equation

Once the equation is entered correctly, graph it. The line should appear on the display. Ensure the visual representation matches the expected slope and y-intercept.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

slope-intercept form
When graphing linear equations, converting your equation to slope-intercept form can make the process easier. The slope-intercept form is written as: y = mx + b Here, 'm' represents the slope of the line, and 'b' is the y-intercept鈥攖he point where the line crosses the y-axis. To convert an equation to this form, you'll need to solve for y and isolate it on one side of the equation. For example, if we start with the equation 3x + 4y = 6:
  • First, subtract 3x from both sides: 4y = -3x + 6.
  • Then, divide every term by 4: y = -戮x + 1陆.
So now, y = -戮x + 1陆 is in slope-intercept form. Here, the slope (m) is -戮, and the y-intercept (b) is 1.5.
graphing calculator
Graphing calculators are essential tools for visualizing linear equations. They can help you quickly see the shape and position of your line on a coordinate plane. Here's how to use one to graph the equation y = -戮x + 1陆:
  • First, turn on the graphing calculator and enter the equation into the graphing function. On most calculators, you'll find this feature in the 'Y=' menu or a similar option.
  • After entering the equation, ensure your graphing calculator is set to the standard viewing window. This will ensure you see a broad and useful range of the graph.
  • The standard viewing window usually ranges from -10 to 10 on both the x-axis and y-axis.
  • Finally, press the graph button. You should see the line graph appear, representing the visual form of the equation y = -戮x + 1陆.
Additionally, many graphing calculators allow you to zoom and adjust the viewing window, providing a closer look at specific parts of the graph.
standard viewing window
The standard viewing window is a preset range for displaying graphs on graphing calculators. It simplifies the process of visualizing equations by providing a standardized, easy-to-read grid. Here's why it's useful:
  • It usually ranges from -10 to 10 along both the x-axis and y-axis.
  • This window size provides a broad and balanced view, making it easier to spot key features of the graph like the slope and y-intercept.
  • When using the standard viewing window, you avoid having to manually adjust the scales, which can be time-consuming and sometimes confusing for beginners.
Using the standard viewing window ensures your graph is displayed in a familiar format, maintaining consistency and allowing for easy comparison with other graphs. If your graph isn鈥檛 fitting well within this window, you can always zoom in or out or adjust the range manually for a better view.

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Most popular questions from this chapter

Solve each problem. Relationship of Measurement Units The function defined by \(f(x)=12 x\) computes the number of inches in \(x\) feet, and the function defined by \(g(x)=5280 x\) computes the number of feet in \(x\) miles. What does \((f \circ g)(x)\) compute?

Show that \((f \circ g)(x)\) is not equivalent to \((g \circ f)(x)\) for $$f(x)=3 x-2 \quad \text { and } \quad g(x)=2 x-3$$

The tables give some selected ordered pairs for functions \(f\) and \(g\). $$\begin{array}{|c|c|c|c|}\hline x & 3 & 4 & 6 \\\\\hline f(x) & 1 & 3 & 9 \\\\\hline\end{array}$$ $$\begin{array}{|c|c|c|c|c|}\hline x & 2 & 7 & 1 & 9 \\\\\hline g(x) & 3 & 6 & 9 & 12 \\\\\hline\end{array}$$ Find each of the following. $$(f \circ g)(2)$$

Decide whether each relation defines \(y\) as a function of \(x\). Give the domain and range. $$y=\sqrt{4 x+1}$$

For certain pairs of functions \(f\) and \(g,(f \circ g)(x)=x\) and \((g \circ f)(x)=x .\) Show that this is true for each pair. $$f(x)=\sqrt[3]{5 x+4}, g(x)=\frac{1}{5} x^{3}-\frac{4}{5}$$
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