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Chapter 0: Problem 82

Use symmetry to sketch the graph of the equation. \(y=1-|x|\)

Short Answer

Expert verified
The graph of the equation \(y = 1 - |x|\) is a V-shaped figure that is symmetric with respect to the y-axis. The point of the V lies at (0,1).

Step by step solution

01

Draw the lines for \(x \geq 0\) and \(x < 0\)

Start by plotting the line \(y = 1 - x\) for all \(x \geq 0\). This line has a slope of -1 and passes through the point (0,1). For \(x < 0\), plot the line \(y = 1 + x\). It has a positive slope of 1 and also passes through the point (0,1).
02

Emphasize the symmetry

Take note of how the two lines meet at the point (0,1) and form a V-shape. This is a reflection symmetry. If you were to fold the graph along the y-axis, the right half would align perfectly with the left half.
03

Final Drawing

Lastly, clean up your sketch to ensure it clearly represents the function. Highlight the point where the lines meet, as this is what causes the symmetry, and label your axes with the function names for clarity.

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Key Concepts

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

Absolute Value
The absolute value of a number represents its distance from zero on the number line, regardless of which direction from zero the number lies. It is always a non-negative value. In mathematical notation, the absolute value of a number 'x' is denoted as |x|. When graphing an absolute value function, you typically see a V-shaped graph.

Looking at the exercise y = 1 - |x|, the absolute value will affect the function's graph by creating two different scenarios: one for when x is positive (or zero) and one for when x is negative. The absolute value will convert all negative values of x into positive ones, effectively reflecting any negative input across the y-axis on the graph.
Symmetry in Graphs
Symmetry in graphs indicates that certain parts of the graph are mirror images of each other. Reflective symmetry about the y-axis is what we're observing in our exercise. When you have a function such as y = 1 - |x|, the graph will show this sort of symmetry; if the function's graph is folded over the y-axis, both halves would match up perfectly.

In the solution we start by analyzing two expressions obtained from the piecewise nature of the absolute value: y = 1 - x for x ≥ 0 and y = 1 + x for x < 0. The graph is symmetrical relative to the y-axis because for every positive x, there's a corresponding negative x that results in the same y-value. This inherent symmetry makes it easier to graph absolute value functions as you can first plot the graph for non-negative inputs and then mirror it across the y-axis for negative inputs.
Piecewise Functions
Piecewise functions are functions that have different expressions or rules for different intervals of the input variable, typically x. Instead of having one formula, a piecewise function has several formulas, each corresponding to a particular part of the domain.

In the context of absolute value functions like y = 1 - |x|, you can think of the function as being composed of two pieces. For x ≥ 0, the function follows the rule y = 1 - x, which is a straight line with a negative slope. For x < 0, the rule changes to y = 1 + x, a line with a positive slope. Graphing a piecewise function involves graphing each 'piece' over its specific interval, and ensuring the graph is seamless where the pieces meet.

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Most popular questions from this chapter

In Exercises 33-38, use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. $$ f(x)=10 $$

In Exercises 69-74, use the functions given by \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$ \left(f^{-1} \circ g^{-1}\right)(1) $$

(a) The sales tax on a purchased item is a function of the selling price. (b) Your score on the next algebra exam is a function of the number of hours you study the night before the exam.

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=x^{2}, \quad x \geq 0 $$

In Exercises 75-78, use the functions given by \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$ f^{-1} \circ g^{-1} $$
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