Mathematics
Grade 12
15 min
Find properties of sine functions
Find properties of sine functions
What you'll learn
- Identify the number that is one more than a given number from 1 to 9 with 80% accuracy.
- Identify the number that is one less than a given number from 2 to 10 with 80% accuracy.
- Solve 'one more' and 'one less' problems using manipulatives (e.g., counters, blocks) for numbers up to 10 in at least 3 out of 4 attempts.
- Explain, using simple language, what 'one more' means by adding one object to a set, in at least 2 out of 3 attempts.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Identify the amplitude, period, phase shift, and vertical shift of a sine function from its equation.
Determine the equation of a sine function given its key properties or its graph.
Calculate the period of a sine function using the formula T = 2π/|B|.
Correctly identify the direction and magnitude of a phase shift by factoring the 'B' coefficient.
State the domain and range of a transformed sine function.
Explain the effect of a reflection across the x-axis and the midline on the sine function's graph and equation.
Have you ever wondered how engineers model the oscillating current in your home's outlets or how animators create realistic wave movements? 🌊 It all starts with understanding the properties of the sine function!
This tuto...
2
Key Concepts & Vocabulary
TermDefinitionExample
AmplitudeThe maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position. It is half the distance between the maximum and minimum values of the function.For y = 3 sin(x), the function oscillates between -3 and 3. The amplitude is |3| = 3.
PeriodThe length of one complete cycle of the wave. It is the horizontal distance after which the function's values begin to repeat.The basic function y = sin(x) has a period of 2π, meaning its graph completes one full cycle from x=0 to x=2π.
Phase ShiftThe horizontal translation (shift left or right) of the sine function from its default position (y = sin(x)).In y = sin(x - π/2), the graph is shifted π/2 units to the right compared to y = sin(x).
Vertical Shift (Midline)...
3
Core Formulas
The General Sine Function
y = A \sin(B(x - C)) + D
This is the standard form used to analyze transformations of the sine function. 'A' controls amplitude and reflection, 'B' controls the period, 'C' controls the phase shift, and 'D' controls the vertical shift.
Period Formula
T = \frac{2\pi}{|B|}
Use this formula to calculate the period (T) of the sine function. The value of B is taken from the general equation y = A sin(B(x - C)) + D. A larger |B| value results in a shorter period (horizontal compression).
Range of a Sine Function
[D - |A|, D + |A|]
The range of a transformed sine function is determined by its vertical shift (D) and amplitude (|A|). The minimum value is D - |A| and the maximum value is D + |A|.
4 more steps in this tutorial
Sign up free to access the complete tutorial with worked examples and practice.
Sign Up Free to ContinueSample Practice Questions
Challenging
A sine function has a maximum point at (π/2, 5) and a subsequent minimum point at (3π/2, -1). Determine a possible equation for this function, as demonstrated in the tutorial.
A.y = 3 sin(x) + 2
B.y = 3 sin(2x) + 2
C.y = 6 sin(x) + 2
D.y = 3 sin(x - π/2) + 2
Challenging
The function f(x) = A sin(B(x-C)) + D has a period of 4π and passes through the point (0, 3). It has a maximum value of 7. Assuming A > 0 and the simplest phase shift, which of the following could be the function?
A.y = 4 sin(x/2 - π) + 3
B.y = 4 sin(1/2(x + π)) + 3
C.y = 7 sin(2(x - π)) + 3
D.y = 4 sin(2x) + 3
Challenging
The graph of y = A sin(Bx) + D is shifted right by π/6 units and up by 1 unit. The resulting graph has the equation y = 2sin(3x - π/2) + 4. What were the original values of A, B, and D?
A.A=2, B=3, D=3
B.A=2, B=3, D=5
C.A=2, B=1, D=3
D.A=4, B=3, D=3
Want to practice and check your answers?
Sign up to access all questions with instant feedback, explanations, and progress tracking.
Start Practicing FreeMore from Trigonometric functions
Mathematics for other grades
Frequently asked questions
What grade level is "Find properties of sine functions"?
Find properties of sine functions is a Grade 12 Mathematics lesson on ExcelOS.
What will I learn in Find properties of sine functions?
You'll be able to: Identify the number that is one more than a given number from 1 to 9 with 80% accuracy; Identify the number that is one less than a given number from 2 to 10 with 80% accuracy; Solve 'one more' and 'one less' problems using….
Is "Find properties of sine functions" free to practice?
Yes. You can read the tutorial preview for free, and signing up for a free ExcelOS account unlocks the full tutorial and all practice questions with instant feedback.
How many practice questions are included with Find properties of sine functions?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.