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Chapter 6: Problem 2

A function is periodic with period 2 and is even. Sketch a possible form of this function.

Short Answer

Expert verified
Sketch a symmetric function mirrored on the y-axis, repeating every 2 units.

Step by step solution

01

Understand Even Functions

An even function is symmetric with respect to the y-axis. This means that for every point \((x, f(x))\), there is a corresponding point \((-x, f(x))\). To sketch, we need to ensure that the function mirrors itself on both sides of the y-axis.
02

Recognize Periodic Functions

A periodic function repeats its values in regular intervals or periods. Since this function has a period of 2, this means that \(f(x+2) = f(x)\) for all \(x\). Our sketch must repeat every 2 units along the x-axis.
03

Combining Even and Periodic Properties

Combine both properties to create a sketch. One simple approach is to start by sketching one period of an even function, such as a parabola \(f(x) = 1 - x^2\) for the interval \(-1 \leq x \leq 1\). This segment must mirror around the y-axis to satisfy the even property. Then, we repeat this segment every 2 units along the x-axis to maintain periodicity.
04

Sketching the Function

Begin by drawing the portion of the function from \(-1\) to \(1\), which could be something symmetric like a downward-facing parabola. Next, duplicate this graph twice: one from \(1\) to \(3\) and another from \(-3\) to \(-1\), ensuring it is symmetrical and repeats every 2 units. Continue this pattern to fill the graph area.
05

Verify the Sketch

Check the sketch to ensure that it repeats every 2 units and remains symmetrical about the y-axis. Adjust if necessary to ensure both periodicity and evenness.

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Key Concepts

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

Symmetry in Even Functions
Even functions are quite special because they possess a unique type of symmetry. This symmetry is all about the y-axis. When a function is 'even,' it means if you take any point on the graph, say \(x, f(x)\), there's another point at \(-x, f(x)\)\ that mirrors it. This creates a reflection over the y-axis.
Think of the y-axis like a vertical mirror. Whatever you see on the right side of the mirror must appear on the left side too. That's the beauty of symmetry in even functions.
  • For instance, the function \(f(x) = 1 - x^2\) is even. If you plot it, the part of the graph from \(-1 \) to \(1\) will look exactly the same on either side of the y-axis.
When sketching such a function, ensure it reflects perfectly across this line.
Graph Sketching Techniques
Graph sketching is an art of visualizing mathematical functions. For even and periodic functions, it starts with understanding their properties. Let's break this down for better understanding.
  • Firstly, identify a basic segment of the function by considering its symmetric nature. For an even function, this segment should reflect across the y-axis. A good example is sketching from \(-1\) to \(1\).
  • Next, to maintain periodicity, repeat this segment at regular intervals, based on the specified period. In this case, the period is \(2\).
  • Essentially, you draw it over one interval and then just copy-paste it across the graph. This gives you a continuous pattern.
By using these steps, you can transform abstract functions into a clear, tangible visual graph.
Understanding Function Periodicity
Periodicity is a fascinating characteristic of some functions. It refers to the pattern that repeats at regular intervals, known as periods. For a function with a period of \(2\), this means that if you pick any number x, the function at \(x + 2\) will be the same as the function at x under all circumstances.
  • Imagine it like a wave or a repeating tile pattern where every 2 units along the x-axis, the function makes a cycle.
  • This helps in predicting the shape of the graph over different intervals, knowing it eventually loops back to the same values.
For sketching, this tells us that once a given segment is drawn, repeating it along the x-axis will maintain the pattern. Over time, this creates beautifully repeating patterns like waves or heartbeats on the graph. Remember, with each complete settiling, the intervals stay consistent, just like clockwork.

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Most popular questions from this chapter

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