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Chapter 1: Problem 45

Identify any intercepts and test for symmetry. Then sketch the graph of the equation. $$y=-3 x+1$$

Short Answer

Expert verified
The graph of the given equation has a y-intercept at y = 1 and an x-intercept at x = 1/3. The graph doesn't show any symmetry about either axis or the origin. The slope of the line is -3.

Step by step solution

01

Intercepts

First, let's find the y-intercept. In the equation \(y = -3x + 1\), the y-intercept is the constant term, so it is y = 1. Next, to find the x-intercept, substitute y with 0 and solve for x. Doing so gives: \(0 = -3x + 1\), which simplifies to \(x = 1/3\) . So, the x-intercept is x = 1/3.
02

Test for symmetry

Now let's see if the equation is symmetric with respect to any of the axes or the origin. For symmetry about the y-axis, replace x with -x. The equation becomes \(y = -3(-x) + 1\), which simplifies to \(y = 3x + 1\). This is not the same as the original equation, so it is not symmetric about the y-axis. When you substitute y with -y, you will get \(-y = -3x + 1\), this is also not the same as the original equation, so it is not symmetric about the x-axis. Similarly replacing both x and y with -x and -y respectively doesn't return the original equation, so it is not symmetric about the origin either.
03

Sketch the graph

With the determined intercepts (x = 1/3, y = 1) and the knowledge that the graph is not symmetric about either axis or the origin, sketch a graph with negative slope (-3). Start at the y-intercept (0,1) and go down 3 units and to the right 1 unit.

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