Math / Trigonometry

Question9. Using electronic graphing tools, graph $y=2 \sin (3[x-4])+5$ and $y=2 \csc (4 x-3)-1$ on the same axes. Find all points of intersection of the two functions between $-\pi$ and $\pi$ , with answers to 2 decimal places. Include an image of the graphs in your response. [ 6 marks]

Studdy Solution

STEP 1

1. We are given two trigonometric functions: $y = 2 \sin(3[x-4]) + 5$ and $y = 2 \csc(4x-3) - 1$ .
2. We need to find the points of intersection of these two functions within the interval $-\pi \leq x \leq \pi$ .
3. The use of electronic graphing tools is necessary to visualize the graphs and find the intersection points accurately.

STEP 2

1. Graph the function $y = 2 \sin(3[x-4]) + 5$ .
2. Graph the function $y = 2 \csc(4x-3) - 1$ .
3. Identify the points of intersection of the two graphs within the specified interval.
4. Calculate the coordinates of the intersection points to two decimal places.
5. Provide an image of the graphs.

STEP 3

Use an electronic graphing tool to graph the function $y = 2 \sin(3[x-4]) + 5$ .
- This function is a sine function with an amplitude of 2, a horizontal shift of 4 units to the right, a vertical shift of 5 units up, and a period determined by the factor 3 inside the sine function.

STEP 4

Use the electronic graphing tool to graph the function $y = 2 \csc(4x-3) - 1$ .
- This function is a cosecant function with an amplitude of 2, a vertical shift of 1 unit down, and a period determined by the factor 4 inside the cosecant function.

STEP 5

Identify the points of intersection of the two graphs within the interval $-\pi \leq x \leq \pi$ .
- Use the graphing tool's intersection feature to find where the two graphs intersect within the specified interval.

STEP 6

Calculate the coordinates of the intersection points to two decimal places.
- Record the $x$ and $y$ coordinates of each intersection point as provided by the graphing tool.

STEP 7

Provide an image of the graphs.
- Capture a screenshot or export the graph from the electronic tool showing both functions and their points of intersection.
Solution:
1. Graph the functions using an electronic graphing tool.
2. Identify and calculate the intersection points within the interval $-\pi \leq x \leq \pi$ .
3. Provide an image of the graphs.
(Note: As a text-based AI, I am unable to generate images directly. Please use a graphing calculator or software like Desmos, GeoGebra, or a graphing calculator to complete this step.)

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Question9. Using electronic graphing tools, graph $y=2 \sin (3[x-4])+5$ and $y=2 \csc (4 x-3)-1$ on the same axes. Find all points of intersection of the two functions between $-\pi$ and $\pi$ , with answers to 2 decimal places. Include an image of the graphs in your response. [ 6 marks]

Studdy Solution

STEP 1

STEP 2

1. Graph the function $y = 2 \sin(3[x-4]) + 5$ .
2. Graph the function $y = 2 \csc(4x-3) - 1$ .
3. Identify the points of intersection of the two graphs within the specified interval.
4. Calculate the coordinates of the intersection points to two decimal places.
5. Provide an image of the graphs.

STEP 3

Use an electronic graphing tool to graph the function $y = 2 \sin(3[x-4]) + 5$ .
- This function is a sine function with an amplitude of 2, a horizontal shift of 4 units to the right, a vertical shift of 5 units up, and a period determined by the factor 3 inside the sine function.

STEP 4

Use the electronic graphing tool to graph the function $y = 2 \csc(4x-3) - 1$ .
- This function is a cosecant function with an amplitude of 2, a vertical shift of 1 unit down, and a period determined by the factor 4 inside the cosecant function.

STEP 5

Identify the points of intersection of the two graphs within the interval $-\pi \leq x \leq \pi$ .
- Use the graphing tool's intersection feature to find where the two graphs intersect within the specified interval.

STEP 6

Calculate the coordinates of the intersection points to two decimal places.
- Record the $x$ and $y$ coordinates of each intersection point as provided by the graphing tool.

STEP 7

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Studdy Solution

STEP 1

STEP 2

STEP 3

STEP 4

Use the electronic graphing tool to graph the function y=2csc⁡(4x−3)−1 y = 2 \csc(4x-3) - 1 y=2csc(4x−3)−1.- This function is a cosecant function with an amplitude of 2, a vertical shift of 1 unit down, and a period determined by the factor 4 inside the cosecant function.

STEP 5

Identify the points of intersection of the two graphs within the interval −π≤x≤π -\pi \leq x \leq \pi −π≤x≤π.- Use the graphing tool's intersection feature to find where the two graphs intersect within the specified interval.

STEP 6

Calculate the coordinates of the intersection points to two decimal places.- Record the x x x and y y y coordinates of each intersection point as provided by the graphing tool.

STEP 7

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Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Use the electronic graphing tool to graph the function $y = 2 \csc(4x-3) - 1$ .
- This function is a cosecant function with an amplitude of 2, a vertical shift of 1 unit down, and a period determined by the factor 4 inside the cosecant function.

Identify the points of intersection of the two graphs within the interval $-\pi \leq x \leq \pi$ .
- Use the graphing tool's intersection feature to find where the two graphs intersect within the specified interval.

Calculate the coordinates of the intersection points to two decimal places.
- Record the $x$ and $y$ coordinates of each intersection point as provided by the graphing tool.