Graphs of Sine & Cosine Functions

Move the knob on each slider and observe what each does. [br]Make sure to take note so that you can answer the questions in the subsequent tasks. [br][br][math]f\left(x\right)=a\cdot\sin\left(b\cdot\left(x-c\right)\right)+d[/math] or [math]f\left(x\right)=a\cdot\cos\left(b\cdot\left(x-c\right)\right)+d[/math]

1.

The [b][color=#0000ff]parameter [i]a[/i][/color][/b] is called the [color=#0000ff][b]amplitude[/b][/color] of either graph. How would you describe the term [i][b][color=#0000ff]amplitude[/color][/b][/i] in your own words?

2.

[br][br]What does the [b][color=#0000ff]amplitude [/color][/b][i][b][color=#0000ff]a[/color][/b] [/i]do to the graph of either function?

As [math]a[/math] increases, the graph of sine and cosine shift horizontally As [math]a[/math] increases, the graph of sine and cosine stretch vertically As [math]a[/math] increases, the graph of sine and cosine shift vertically As [math]a[/math] increases, the graph of sine and cosine stretch horizontally

3.

The [b]period [/b]of a function is the length of the largest interval in its domain before it repeats itself.The sine and cosine functions are said to be periodic functions because they have periods. To learn more about [i]periodic functions[/i], [url=https://www.geogebra.org/m/QRmStjFg]click here[/url].[br][br]What is the period of both parent functions [math]f\left(x\right)=sin\left(x\right)[/math] or [math]f\left(x\right)=cos\left(x\right)[/math]? [br](If you need a hint, [url=https://www.geogebra.org/m/dj8jwyNV]click here[/url].) [br][br][br]

[math]\pi[/math] 4[math]\pi[/math] 2[math]\pi[/math] 3[math]\pi[/math]

4.

The [b][color=#38761d]parameter [i]b[/i][/color][/b] gives the [color=#38761d][b]period[/b][/color] of the function by the formula [math]\frac{2\pi}{b}[/math]. What happens to the wave if b is between 0 and 1?

The graph of the function contracts vertically. The graph of the function contracts horizontally. The graph of the function stretches vertically. The graph of the function stretches horizontally.

5.

How does the [b][color=#ff00ff]parameter [i]c[/i][/color][/b] affect the graph of either parent function? [br][br]How does the graph of this parent function change as[b][color=#ff00ff] [i]c[/i] increases[/color][/b][br]

The graph of the parent function shifts vertically up The graph of the parent function shifts vertically down The graph of the parent function shifts horizontally to the left The graph of the parent function shifts horizontally to the right

6.

How does the [b][color=#980000]parameter [i]d[/i][/color][/b] affect the graph of either parent function? [br][br]How does the graph of this parent function change as[b][color=#980000] [i]d [/i]increases[/color][/b]?

The graph of the parent function shifts vertically up The graph of the parent function shifts horizontally to the right The graph of the parent function shifts horizontally to the left The graph of the parent function shifts vertically down

Graphs of Sine & Cosine Functions

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7. Move the sliders to create a sine graph with amplitude: 3 and period 2 (or close to it)

8. Move the sliders to create a cosine graph with amplitude: 2 and period 3 (or close to it) shifted down by 1.

Information: Graphs of Sine & Cosine Functions