Math  /  Trigonometry

QuestionQuestion 1 of 3, Step 2 of 2 TRINITEE NEWMAN 8/6 Correct
Graph the following function: y=352cot(x)y=3-\frac{5}{2} \cot (x)
Step 2 of 2: Determine how the general shape of the graph, chosen in the previous step, would be shifted, stretched, and reflected for the given function. Graph the results on the axes provided.
Answer Keypac Keyboard Shortch xx-Axis Reflection nenect graph across xx-axis
Shift Graph Vertically UpU_{p} Down None Shift Graph Horizontally (Phase Shift) Left Right None
Stretch/Compress Graph Vertically Yes No
Stretch/Compress Graph Horizontally (Period) Yes No Enable Zoom/Pan Submit Answer - 2024 Hawkes Learning D=14D=14

Studdy Solution

STEP 1

What is this asking? We need to describe how the graph of y=352cot(x)y = 3 - \frac{5}{2} \cot(x) is transformed compared to the basic cotangent graph, y=cot(x)y = \cot(x), and then sketch the graph. Watch out! Don't mix up the vertical shift and the vertical stretch/compression.
Also, remember that the *negative* sign in front of the cotangent function causes a reflection.

STEP 2

1. Analyze the Vertical Shift
2. Analyze the Reflection and Vertical Stretch
3. Analyze the Horizontal Stretch/Compression
4. Sketch the Graph

STEP 3

In the function y=352cot(x)y = 3 - \frac{5}{2} \cot(x), the **+3 +3 ** shifts the graph **upward** by **3 units**.
It's like taking the entire cotangent graph and lifting it!

STEP 4

The coefficient **52- \frac{5}{2}** in front of the cotangent function does two things.
First, the **negative sign** reflects the graph across the x-axis.
It's like flipping it upside down!

STEP 5

Second, the **absolute value** of the coefficient, which is 52\frac{5}{2} or **2.5**, stretches the graph vertically.
This makes the graph taller, increasing the distance between the peaks and valleys.

STEP 6

Since there's no number multiplied directly with the xx inside the cotangent function (like cot(2x)\cot(2x) or cot(12x)\cot(\frac{1}{2}x)), there's *no* horizontal stretch or compression.
The period of the function remains the same as the basic cotangent function, which is π\pi.

STEP 7

Start by sketching the basic cotangent graph, y=cot(x)y = \cot(x).
Remember, it has vertical asymptotes at x=kπx = k\pi, where kk is an integer.

STEP 8

Now, **reflect** this graph across the x-axis because of the negative sign in our function.

STEP 9

Next, **stretch** the reflected graph vertically by a factor of **2.5**.
This makes the graph appear steeper.

STEP 10

Finally, **shift** the entire graph **upward** by **3 units** due to the +3+3 in our function.
This moves all points on the graph 3 units up.

STEP 11

The graph of y=352cot(x)y = 3 - \frac{5}{2} \cot(x) is the graph of y=cot(x)y = \cot(x) reflected across the x-axis, vertically stretched by a factor of 2.5, and shifted upward 3 units.
The period remains π\pi.

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