
To find: Explain how the amplitude and period of a sinusoidal graph are used to establish the scale on each coordinate axis:

Answer to Problem 39AYU
Solution:
Sinusoidal functions produce graphs that look like waves, and any portion of the curve can be translated onto another portion of the curve. Circular motion and pendulum movement are excellent “real life” examples of sinusoidal relationships.
A sinusoidal function is a function that is like a function in the sense that the function can be produced by shifting, stretching or compressing the function.
You can graph sinusoidal functions using your knowledge of transformations.
There are 5 transformations that can affect trig functions:
- horizontal stretch (HS) alters the period
- vertical stretch (VS) alters the amplitude
- horizontal translation (HT) causes a phase shift relocating our initial point
- vertical translation (VT) relocates the sinusoidal axis
- reflection (± )changes the pattern of our 5 key points
Explanation of Solution
Sinusoidal functions produce graphs that look like waves, and any portion of the curve can be translated onto another portion of the curve. Circular motion and pendulum movement are excellent “real life” examples of sinusoidal relationships.
A sinusoidal function is a function that is like a function in the sense that the function can be produced by shifting, stretching or compressing the function.
You can graph sinusoidal functions using your knowledge of transformations.
There are 5 transformations that can affect trig functions:
- horizontal stretch (HS) alters the period
- vertical stretch (VS) alters the amplitude
- horizontal translation (HT) causes a phase shift relocating our initial point
- vertical translation (VT) relocates the sinusoidal axis
- reflection (± )changes the pattern of our 5 key points
Some basic vocabulary should be observed as shown in the diagram:
Period: a complete cycle of the graph where the graph starts to repeat.
Sinusoidal axis: horizontal line halfway between the
Amplitude: vertical distance from the sinusoidal axis to the maximum or minimum
Local maximum: the tops of each “wave”, all equal in value
The relations involving or can be expressed in the form:
- Determine the position of the sinusoidal axis.
This is the Vertical Translation or “D” in the formula.
- Determine the amplitude of the function.
This is the Vertical Stretch or “A” in the formula.
- Determine the period of the graph for the function.
This is the Horizontal Stretch or “B” in the formula.
- Determine the phase shift.
This is the Horizontal Translation or “C” in the formula.
Graphing a Sinusoidal Function
The procedure for graphing sinusoidal functions is similar to quadratic functions.
or
Here A is the amplitude function. We know the maximum and minimum value of and is . Therefore takes value from to .
Both and have the domain is .
Therefore takes values are .
Chapter 6 Solutions
Precalculus Enhanced with Graphing Utilities
Additional Math Textbook Solutions
A First Course in Probability (10th Edition)
Elementary Statistics
Introductory Statistics
Algebra and Trigonometry (6th Edition)
Elementary Statistics (13th Edition)
College Algebra (7th Edition)
- u, v and w are three coplanar vectors: ⚫ w has a magnitude of 10 and points along the positive x-axis ⚫ v has a magnitude of 3 and makes an angle of 58 degrees to the positive x- axis ⚫ u has a magnitude of 5 and makes an angle of 119 degrees to the positive x- axis ⚫ vector v is located in between u and w a) Draw a diagram of the three vectors placed tail-to-tail at the origin of an x-y plane. b) If possible, find w × (ū+v) Support your answer mathematically or a with a written explanation. c) If possible, find v. (ū⋅w) Support your answer mathematically or a with a written explanation. d) If possible, find u. (vxw) Support your answer mathematically or a with a written explanation. Note: in this question you can work with the vectors in geometric form or convert them to algebraic vectors.arrow_forwardQuestion 3 (6 points) u, v and w are three coplanar vectors: ⚫ w has a magnitude of 10 and points along the positive x-axis ⚫ v has a magnitude of 3 and makes an angle of 58 degrees to the positive x- axis ⚫ u has a magnitude of 5 and makes an angle of 119 degrees to the positive x- axis ⚫ vector v is located in between u and w a) Draw a diagram of the three vectors placed tail-to-tail at the origin of an x-y plane. b) If possible, find w × (u + v) Support your answer mathematically or a with a written explanation. c) If possible, find v. (ū⋅ w) Support your answer mathematically or a with a written explanation. d) If possible, find u (v × w) Support your answer mathematically or a with a written explanation. Note: in this question you can work with the vectors in geometric form or convert them to algebraic vectors.arrow_forwardK Find all values x = a where the function is discontinuous. For each value of x, give the limit of the function as x approaches a. Be sure to note when the limit doesn't exist. x-7 p(x) = X-7 Select the correct choice below and, if necessary, fill in the answer box(es) within your choice. (Use a comma to separate answers as needed.) OA. f is discontinuous at the single value x = OB. f is discontinuous at the single value x= OC. f is discontinuous at the two values x = OD. f is discontinuous at the two values x = The limit is The limit does not exist and is not co or - ∞. The limit for the smaller value is The limit for the larger value is The limit for the smaller value is The limit for the larger value does not exist and is not c∞ or -arrow_forward
- K x3 +216 complete the table and use the results to find lim k(x). If k(x) = X+6 X-6 X -6.1 -6.01 - 6.001 - 5.999 - 5.99 -5.9 k(x) Complete the table. X -6.1 -6.01 - 6.001 - 5.999 - 5.99 - 5.9 k(x) (Round to three decimal places as needed.) Find the limit. Select the correct choice below and, if necessary, fill in the answer box within your choice.arrow_forwardSketch the slope field that represents the differential equation. × Clear Undo Redo y ४|० || 33 dy dxarrow_forwardSketch the slope field that represents the differential equation. × Clear Undo Redo dy 33 dx = -y "arrow_forward
- Math Test 3 3 x³+y³ = Ꭹ = 9 2 2 x²+y² = 5 x+y=?arrow_forwardFor each of the following series, determine whether the absolute convergence series test determines absolute convergence or fails. For the ¿th series, if the test is inconclusive then let Mi = 4, while if the test determines absolute convergence let Mi 1 : 2: ∞ Σ(−1)"+¹ sin(2n); n=1 Σ n=1 Σ ((−1)”. COS n² 3+2n4 3: (+ 4: 5 : n=1 ∞ n 2+5n3 ПП n² 2 5+2n3 пп n² Σ(+)+ n=1 ∞ n=1 COS 4 2 3+8n3 П ηπ n- (−1)+1 sin (+727) 5 + 2m³ 4 = 8. Then the value of cos(M₁) + cos(2M2) + cos(3M3) + sin(2M) + sin(M5) is -0.027 -0.621 -1.794 -1.132 -1.498 -4.355 -2.000 2.716arrow_forwardi need help with this question i tried by myself and so i am uploadding the question to be quided with step by step solution and please do not use chat gpt i am trying to learn thank you.arrow_forward
- i need help with this question i tried by myself and so i am uploadding the question to be quided with step by step solution and please do not use chat gpt i am trying to learn thank you.arrow_forward1. 3 2 fx=14x²-15x²-9x- 2arrow_forwardNo it is not a graded assignment, its a review question but i only have the final answer not the working or explanationarrow_forward
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning





