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Chapter 1: Problem 53

Identify any intercepts and test for symmetry. Then sketch the graph of the equation. $$y=|x-6|$$

Short Answer

Expert verified
The x-intercept and y-intercept of the function are both at 6. The function is not symmetric about the y-axis or the line \(y = x\). The graph is a V-shape that opens upwards and is shifted 6 units to the right along the x-axis.

Step by step solution

01

Finding the x- and y-intercepts

The x-intercept is found by setting \(y\) equal to zero and solving for \(x\), so: 0= |x - 6|. The solution is \(x = 6\). For the y-intercept, set \(x\) equal to zero and solve for \(y\), thus: \(y = |0 - 6|\), which results in \(y=6\).
02

Testing for Symmetry

Let's test for symmetry about the y-axis. Replace \(x\) with \(-x\) in the equation \(y = |x - 6|\), resulting in \(y = |-x - 6|\). This is not equal to the original function, so the graph isn't symmetric about the y-axis. Now, let's test for symmetry about the line \(y = x\). Swap \(x\) and \(y\) in the function to get \(x = |y-6|\). This is also not equal to the original function, so the graph isn't symmetric about the line \(y = x\).
03

Sketching the graph

Plot the intercepts on the graph; they're both at (6, 0) and (0,6). The graph follows the shape of \(y = |x|\), shifted 6 units to the right along the x-axis. Hence it's a V-shape that opens upwards, centered at the point (6, 0).

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