Module 4: Lesson 4.05 - Graphs of Sine and Cosine Functions
What does period determine?
Horizontal stretching/shrinking of the curve
Cosine Curves: when does a reflection over y-axis occur?
If the coeff of x is negative
What is the Standard Form Equation of Sine?
y = d + a sin(bx - c)
What is the Standard Form Equation of Cosine?
y = d a cos(bx-c)
Sine Curve
Graphs of the sine function
What does amplitude determine?
Vertical stretching/shrinking of the curve & the range such that -a <= y <= a
Sine Curves: what do the y-values of the max & min points reflect?
A change in amplitude (-a <= y <= a)
What does the Phase Shift determine?
A horizontal shift of the curve along the x-axis
Sine Curves: how do you find the y-intercept?
Evaluate the function at x = 0
What are four basic points that help to draw basic sine/cosine functions & graph transformations along those curves?
Max & min points, & x-axis intercepts & y-axis intercepts
Sine Curves: unless the curve is reflected, where does the max & min always occur?
Max: btwn the 1st & 2nd x-intercepts Min: btwn the 2nd & 3rd x-intercepts
Where does a sine curve always begin, middle, and end?
On the x-axis
Cosine Curves: what does a negative 'a' value indicate?
Reflection over x-axis
What are the three characteristics of a Sine Curve?
Repeats infinitely in both directions along x-axis Period is 2π Range: -1 to +1
For the graphs y = a sin(bx - c) & y = a cos(bx - c), how can you determine the left & right endpoints of a one-cycle interval?
Solve: bx - c = 0 bx - c = 2π
With the Phase Shift, what is the amount of shift equal to?
The distance c/b
How does the characteristics of a Cosine Curve compare to those of a Sine Curve?
They're the same, but Cosine Curve is offset to the left by π/2 units
According to the standard forms of the equations for sine and cosine, what is the period?
Value of '2π/b'
According to the standard forms of the equations for sine and cosine, what is the amplitude?
Value of 'a', or |a|
According to the standard forms of the equations for sine and cosine, what is the Phase Shift?
Value of 'c', or c/b
According to the standard forms of the equations for sine and cosine, what is the Vertical Shift?
Value of 'd'
What are the rules for positive b (period) values?
Values < 1 stretch 2π Values > 1 shrink the period to less than 2π
What are the rules for negative b (period) values?
Values are such that: sin (-x) = -sin(x) cos(-x) = cos(x)
