Home > Trigonometry > Translating Sine and Cosine Functions

How do you graph #y=3sin(x-pi/4)+2#?

Answer 1

This a scaling and translation of the basic #y=sin x# graph.

It is being scaled by a factor of three, then translated (moved) 2 up and #pi/4# to the right.

First look at the graph of sin:

Points to note are that it goes through the origin (0,0) and goes up to 1, and down to -1. It crosses the #x# axis at #pi#, #2pi#, #3pi#.... and #-pi#, #-2pi# etc etc.

I have used the Desmos graphing tool to produce these graphs. I'm afraid I've not been able to change the #x# axis to show #pi# rather than numerical values but I hope you'll be able to see that it is crossing approximately every 3.14 (#pi#).

The first step I've done is to move the graph #pi/4# to the right - if you compare this graph and the one above, you can see it's just moved over slightly.:

Next, I've scaled it by a factor of 3, which means that it instead of going between 1 and -1, it will be between 3 and -3. Please note that the graph still crosses the #x# axis in the same places:

Finally, the graph needs to be moved 2 up and this gives you the final answer:

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Zoe Austin

To graph the function (y = 3\sin(x - \frac{\pi}{4}) + 2), you can follow these steps:

Identify the key components: The function is in the form (y = A\sin(Bx - C) + D), where (A) is the amplitude, (B) is the frequency, (C) is the phase shift, and (D) is the vertical shift.
Amplitude: (A = 3). This means the graph will oscillate between (3) and (-3).
Frequency and Period: For (y = \sin(x)), the period is (2\pi). The period of (y = \sin(Bx)) is (\frac{2\pi}{B}). In this case, (B = 1), so the period remains (2\pi).
Phase Shift: (C = \frac{\pi}{4}). This indicates a shift to the right by (\frac{\pi}{4}).
Vertical Shift: (D = 2). This means the entire graph will shift up by (2) units.
Sketch the graph: Start with the basic sine function (y = \sin(x)), then apply the transformations:
- Increase the amplitude to (3) (oscillates between (3) and (-3)).
- Shift the graph to the right by (\frac{\pi}{4}).
- Shift the entire graph up by (2) units.
Plot key points: Choose points at intervals of (\frac{\pi}{4}) or (\frac{\pi}{2}) to plot the graph accurately.
Draw the graph: Connect the plotted points smoothly to form the graph of (y = 3\sin(x - \frac{\pi}{4}) + 2).

Following these steps will enable you to accurately graph the function (y = 3\sin(x - \frac{\pi}{4}) + 2).

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.