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Chapter 1: Problem 64

Explain how to graph an equation in the rectangular coordinate system.

Short Answer

Expert verified
To graph an equation in the rectangular coordinate system, ascertain the function's form, then create a graph grid. Next, plot points that satisfy the function, connect them appropriately, and finally analyze the graph's behavior. The graph visualization helps in understanding the relationship between the variables involved.

Step by step solution

01

Analyze the function

First, it's important to understand the equation to be graphed. If given a function in the form of \(y = f(x)\) - a very common representation - the goal will be to find pairs of \(x\) and \(y\) that satisfy this equation.
02

Create the graph grid

Each graph should begin with creating a grid, which has a horizontal axis \(x\) and a vertical axis \(y\). Label these axes accordingly.
03

Plot the points on the graph

Next, begin plotting points that solve the equation. Start with simple solutions, like \(x = 0, -1, 1, -2, 2\), and so on, and for each \(x\), calculate the corresponding \(y\) value. Each pair \((x, y)\) represents a point on the Cartesian plane that can be plotted.
04

Connect the points

After having a collection of points on the graph, connect these points using a smooth curve or line, based on the nature of the function, to create the graph of the equation.
05

Analyze the graph

Once the graph is drawn, it can be used for visualizing and understanding the relationship between the two variables. How the graph looks can tell us if the relation is linear, quadratic, exponential, etc.

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Most popular questions from this chapter

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=\frac{1}{x}$$

Use a graphing utility to graph each function. Use \(a[-5,5,1]\) by [-5,5,1] viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$f(x)=x^{\frac{1}{3}}(x-4)$$

What is a piecewise function?

Suppose that \(h(x)=\frac{f(x)}{g(x)} .\) The function \(f\) can be even,odd, or neither. The same is true for the function \(g .\) a. Under what conditions is \(h\) definitely an even function? b. Under what conditions is \(h \quad\) definitely an odd function?

Explaining the Concepts: If equations for \(f\) and \(g\) are given, explain how to find \(f-g .\)
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