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Chapter 1: Problem 54

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

Short Answer

Expert verified
The graph of \(y = 1-|x|\) has intercepts at \(x = 1\), \(x = -1\) and \(y = 1\). The graph is symmetric about the y-axis and is a v-shaped graph pointing downwards, peaking at y=1.

Step by step solution

01

Identify the intercepts

Identifying the intercepts means finding the values of x and y where the graph crosses the x-axis and y-axis. For the y-intercept, set \(x = 0\) and solve for y. For the x-intercept(s), set \(y = 0\) and solve for x. In this case, the y-intercept is \(y = 1-|0| = 1\). Solving for the x-intercept gives \(x = 1\) and \(x = -1\).
02

Test for symmetry

To check if the graph is symmetric about the y-axis, replace \(x\) with \(-x\) in the equation and simplify. If the equation remains the same, the graph is symmetric about the y-axis. In this case, replacing \(x\) with \(-x\) in the equation yields \(y = 1-|-x|\), which is equal to the original equation. Therefore, the graph is symmetric about the y-axis.
03

Sketch the graph

Now to sketch the graph, mark the intercepts on the coordinate plane. As the graph is symmetric about the y-axis, draw a line of symmetry at \(x = 0\). The graph of \(y = 1-|x|\) will be a v-shaped graph pointing downwards centred at the y-intercept (which is also the vertex) and forming equal intercepts \(x = 1\) and \(x = -1\) on the x-axis.

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