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)