Problem 26
Question
Draw the graphs of $$ f(x)=\max \left\\{\sin \left(x-\frac{\pi}{4}\right), \sin \left(x+\frac{\pi}{4}\right)\right\\} $$
Step-by-Step Solution
Verified Answer
The graph of the function \(f(x) = \max \left\{\sin \left(x-\frac{\pi}{4}\right), \sin \left(x+\frac{\pi}{4}\right)\right\}\) looks like overlapping waves, where at each x-point the maximum y-value of either sine function is taken.
1Step 1: Graphing Shifted Sine Functions
Start by graphing the two sine functions separately. First, graph \(y=\sin \left(x-\frac{\pi}{4}\right)\). This is a sine function shifted \(\frac{\pi}{4}\) units to the right. Next, graph \(y=\sin \left(x+\frac{\pi}{4}\right)\). This function is shifted \(\frac{\pi}{4}\) units to the left. Identifying the key points for the sine function (i.e., maximums, minimums, zeros) might help to graph it correctly.
2Step 2: Combine the Graphs
To combine the graphs into a single function \(f(x)=\max \left\{\sin \left(x-\frac{\pi}{4}\right), \sin \left(x+\frac{\pi}{4}\right)\right\}\), for every value of x, pick the maximum y value from either of the two graphs. This will be the y-value for your final graph at that x-point.
3Step 3: Finalize the Graph
Do this for a range of x-values to get a complete picture of the final graph. Double check if the graph is periodic with period \(2\pi\) (like any sine function), and identify any particular points of interest, such as maximums, minimums and points of non-differentiability (effective 'corners' in the graph).
Key Concepts
Shifted Sine FunctionMaximum FunctionGraphing Techniques
Shifted Sine Function
When dealing with trigonometric graphs, the shifted sine function represents a common transformation. A sine function naturally oscillates between -1 and 1. By shifting it horizontally, we can explore new positions of this wave within the coordinate plane.
To understand this more deeply, think about the basic sine function, \( y = \sin(x) \). To create a shifted version, we either add or subtract a value from \( x \).
Let's look at the function \( y = \sin(x - \frac{\pi}{4}) \). Here, the sine wave is shifted to the right by \( \frac{\pi}{4} \). This means each key point of the sine wave moves rightward by this amount.
Conversely, consider \( y = \sin(x + \frac{\pi}{4}) \). Here, the shift is to the left by \( \frac{\pi}{4} \).
To understand this more deeply, think about the basic sine function, \( y = \sin(x) \). To create a shifted version, we either add or subtract a value from \( x \).
Let's look at the function \( y = \sin(x - \frac{\pi}{4}) \). Here, the sine wave is shifted to the right by \( \frac{\pi}{4} \). This means each key point of the sine wave moves rightward by this amount.
Conversely, consider \( y = \sin(x + \frac{\pi}{4}) \). Here, the shift is to the left by \( \frac{\pi}{4} \).
- Right-shift: Subtract from \( x \) inside \( \sin(x) \).
- Left-shift: Add to \( x \) inside \( \sin(x) \).
Maximum Function
The concept of a maximum function is pivotal, especially when combining multiple functions into one graph, like in our exercise. The maximum function selects the highest value among different functions at every point along the x-axis.
In mathematical notation, the maximum of two functions \( g(x) \) and \( h(x) \) is represented as \( f(x) = \max\{g(x), h(x)\} \). This means, at each point \( x \), the graph of \( f(x) \) will display whichever of \( g(x) \) or \( h(x) \) has the higher y-value at that x.
Considering our specific example, we evaluate the functions \( \sin(x - \frac{\pi}{4}) \) and \( \sin(x + \frac{\pi}{4}) \) for each \( x \) to find which has the larger y-value. This value becomes the y-coordinate for our graph \( f(x) \).
In mathematical notation, the maximum of two functions \( g(x) \) and \( h(x) \) is represented as \( f(x) = \max\{g(x), h(x)\} \). This means, at each point \( x \), the graph of \( f(x) \) will display whichever of \( g(x) \) or \( h(x) \) has the higher y-value at that x.
Considering our specific example, we evaluate the functions \( \sin(x - \frac{\pi}{4}) \) and \( \sin(x + \frac{\pi}{4}) \) for each \( x \) to find which has the larger y-value. This value becomes the y-coordinate for our graph \( f(x) \).
- This operation is piece-wise in nature.
- Sometimes leads to 'corners' or points of non-differentiability.
Graphing Techniques
Graphing techniques are essential tools for visualizing trigonometric functions and understanding their behavior. It requires knowing how to plot basic sine waves and apply transformations like shifting or stretching.
Start by identifying key points in the periodic cycle of a sine function: maximums, minimums, and zeros. For a comprehensive graph, choose points where these key characteristics change, typically around multiples of \( \frac{\pi}{2} \).
When combining graphs as seen in the maximum function scenario, the technique involves overlaying two functions and choosing the highest points.
Here are some graphing tips:
Start by identifying key points in the periodic cycle of a sine function: maximums, minimums, and zeros. For a comprehensive graph, choose points where these key characteristics change, typically around multiples of \( \frac{\pi}{2} \).
When combining graphs as seen in the maximum function scenario, the technique involves overlaying two functions and choosing the highest points.
Here are some graphing tips:
- Sketch each sine function separately before combining.
- Use graph paper or digital tools for precision.
- Label axes and scales adequately.
- Note periodicity, which for a sine function is \( 2\pi \).
Other exercises in this chapter
Problem 24
Draw the graphs of $$ f(x)=\sin ^{2} x-1 $$
View solution Problem 25
Draw the graphs of $$ f(x)=1-\sin ^{2} x $$
View solution Problem 26
Draw the graphs of $$ f(x)=\sin ^{2} x+\cos ^{2} x $$
View solution Problem 27
Draw the graphs of $$ f(x)=\tan x+\tan |x| $$
View solution