Mathematics
Grade 12
15 min
Write equations of cosine functions from graphs
Write equations of cosine functions from graphs
What you'll learn
- Identify numbers 0 to 10 on a number line with 100% accuracy.
- Point to the number that is one more than a given number (up to 9) on a number line with 80% accuracy.
- Use a number line to count forward from a given number (up to 5) by one, correctly identifying the next two numbers with 75% accuracy.
- Determine which of two numbers (between 1 and 10) is bigger by using a number line, and correctly answer 'Which is bigger?' at least 3 out of 4 times.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Identify the amplitude, period, phase shift, and vertical shift of a cosine function from its graph.
Calculate the 'b' value for the cosine equation using the period found from the graph.
Determine the vertical shift (k) by finding the midline of the function.
Determine the amplitude (a) and its sign (positive or negative) based on the graph's orientation.
Identify the phase shift (h) by locating a maximum point on the graph.
Synthesize the identified parameters (a, b, h, k) to construct a complete and accurate cosine equation.
Write multiple equivalent cosine equations for a single graph by choosing different phase shifts.
Ever wondered how engineers model the cyclical motion of a piston or how scientists predict ocean tides? 🌊 They tra...
2
Key Concepts & Vocabulary
TermDefinitionExample
Midline (Vertical Shift, k)The horizontal line that passes exactly halfway between the function's maximum and minimum values. The y-value of this line is the vertical shift, 'k'.If a function's maximum is y=8 and its minimum is y=2, the midline is y = (8+2)/2 = 5. So, k=5.
Amplitude (a)The distance from the midline to either the maximum or minimum value of the function. It represents the 'height' of the wave. The value of 'a' is positive if the function reaches a maximum at its starting point (after the phase shift) and negative if it reaches a minimum.For a function with a maximum of 8 and a minimum of 2, the amplitude is |a| = (8-2)/2 = 3.
Period (P)The length of the smallest horizontal interval over which the function's...
3
Core Formulas
The Standard Cosine Equation
y = a \cos(b(x - h)) + k
This is the target form for our equation. 'a' is the amplitude and reflection, 'b' relates to the period, 'h' is the phase (horizontal) shift, and 'k' is the vertical shift (midline).
Midline and Amplitude Formulas
k = \frac{y_{max} + y_{min}}{2} \quad \text{and} \quad |a| = \frac{y_{max} - y_{min}}{2}
Use the maximum and minimum y-values from the graph to directly calculate the vertical shift (k) and the absolute value of the amplitude (a).
Period and 'b' Value Formula
P = \frac{2\pi}{|b|} \quad \Longleftrightarrow \quad |b| = \frac{2\pi}{P}
After measuring the period (P) from the graph (the length of one full cycle), use this formula to calculate the value of �...
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Sign Up Free to ContinueSample Practice Questions
Easy
A cosine function has a maximum y-value of 8 and a minimum y-value of 2. What is the equation of the midline?
A.y = 3
B.y = 6
C.y = 5
D.y = 2
Easy
For a function with a maximum y-value of 8 and a minimum y-value of 2, what is the amplitude?
A.6
B.5
C.3
D.10
Easy
If a cosine graph reaches a maximum at x=0 and has not been reflected, what can be concluded about the value of 'a' in the equation y = a cos(b(x - h)) + k?
A.a < 0
B.a = 0
C.a > 0
D.a = k
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What grade level is "Write equations of cosine functions from graphs"?
Write equations of cosine functions from graphs is a Grade 12 Mathematics lesson on ExcelOS.
What will I learn in Write equations of cosine functions from graphs?
You'll be able to: Identify numbers 0 to 10 on a number line with 100% accuracy; Point to the number that is one more than a given number (up to 9) on a number line with 80% accuracy; Use a number line to count forward from a given number (up to….
Is "Write equations of cosine functions from graphs" free to practice?
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How many practice questions are included with Write equations of cosine functions from graphs?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.