Grade 11

Advanced

10 min

Lesson 10 of 12

Graphing Trigonometric Functions

Not Started

Visualize periodic behavior by graphing sine and cosine waves with transformations.

How do radio waves, heartbeats, and ocean tides share a mathematical signature? They all dance to the rhythm of the sine wave, a pattern that repeats perfectly forever.

Learning Objectives

1.By the end of this lesson, you will be able to identify and calculate amplitude, period, phase shift, and vertical displacement.
2.You'll understand how to sketch transformed sine and cosine graphs using the five-point method.
3.You will be able to derive a trigonometric equation from a visual graph.

The Anatomy of a Wave

To master graphing, we must first decode the general equation for a transformed trigonometric function:

y = a \sin(b(x - h)) + k

(The same applies to cosine). Each variable acts as a 'dial' that changes the wave's shape. The Amplitude (

|a|

) measures the height from the center to the peak. The Midline (

y = k

) is the horizontal axis the wave oscillates around. If

a

is negative, the graph reflects across the midline. Understanding these vertical transformations is the first step in visualizing how a function moves away from its parent state,

y = \sin(x)

Identify the amplitude and midline of the function:

y = -3 \sin(x) + 5

1. Locate the value of $a$ . Here, $a = -3$ . The amplitude is the absolute value, so $|-3| = 3$ . 2. Locate the value of $k$ . Here, $k = 5$ . The midline is the horizontal line $y = 5$ . 3. Note the negative sign: The wave will start by going down from the midline instead of up.

Quick Check

If a cosine wave has a maximum value of 10 and a minimum value of 2, what is its amplitude and midline?

The midline is the average of the max and min; the amplitude is the distance from the midline to either extreme.

Answer

The amplitude is 4 and the midline is y = 6.

The Rhythm: Period and Phase Shift

While $a$ and $k$ affect height, $b$ and $h$ affect the 'horizontal' behavior. The Period is the distance required for one full cycle. It is calculated as $P = \frac{2\pi}{|b|}$ . If $b > 1$ , the wave compresses; if $0 < b < 1$ , it stretches. The Phase Shift ( $h$ ) represents a horizontal slide. Crucially, in the form $b(x - h)$ , if you see $(x - rac{\pi}{2})$ , the graph shifts right. If you see $(x + rac{\pi}{2})$ , it shifts left. Mastering the relationship between $b$ and the period is essential for mapping the x-axis correctly.

Find the period and phase shift of

y = \cos(2x - \pi)

1. Factor out the $b$ value to see the shift clearly: $y = \cos(2(x - rac{\pi}{2}))$ . 2. Calculate the period: $P = \frac{2\pi}{2} = \pi$ . 3. Identify the phase shift: Since it is $(x - rac{\pi}{2})$ , the shift is $rac{\pi}{2}$ units to the right.

Quick Check

A function has a $b$ value of $\frac{1}{2}$ . What is its period?

Use the formula

P = \frac{2\pi}{b}

Answer

The period is $4\pi$ .

The Five-Point Method

To sketch any wave accurately, use the Five-Point Method. Every basic cycle of sine or cosine has five key points: the start, the first quarter, the midpoint, the third quarter, and the end. 1. Find the start (Phase Shift $h$ ). 2. Find the end (Phase Shift + Period). 3. Divide the interval into four equal parts by finding the midpoint and the quarter-points. 4. Evaluate the function at these five x-values to find the corresponding y-values (max, min, or midline). This systematic approach ensures your graph is perfectly scaled every time.

Sketch one cycle of $y = 2 \sin(4(x - rac{\pi}{4})) + 1$ .

1. Parameters: Amp = 2, Midline $y=1$ , Period = $\frac{2\pi}{4} = rac{\pi}{2}$ , Phase Shift = $rac{\pi}{4}$ (Right). 2. X-Interval: Starts at $rac{\pi}{4}$ . Ends at $rac{\pi}{4} + rac{\pi}{2} = rac{3\pi}{4}$ . 3. Key Points: - Start: $(rac{\pi}{4}, 1)$ - Quarter: $(rac{5\pi}{16}, 3)$ [Max] - Mid: $(rac{3\pi}{8}, 1)$ - 3rd Quarter: $(rac{7\pi}{16}, -1)$ [Min] - End: $(rac{3\pi}{4}, 1)$

Key Takeaways

1The standard form $y = a \sin(b(x - h)) + k$ allows you to identify all transformations at a glance.
2The period is always $2\pi$ divided by the horizontal frequency $b$ .
3Always factor out $b$ before identifying the phase shift $h$ .
4The five-point method involves dividing the period into four equal increments to find peaks, valleys, and midline intercepts.

Quick Check

Multiple Choice

What is the period of the function $y = 5 \cos(\frac{\pi}{3}x)$ ?

Multiple Choice

In the equation $y = 2 \sin(x + \pi) - 4$ , what is the vertical shift?

True/False

The amplitude of $y = -4 \cos(x)$ is $-4$ .

What's Next?

Review Tomorrow

In 24 hours, try to write down the formula for the period and explain how to find the five key points of a sine graph from memory.

Practice Activity

Find a real-world periodic data set (like monthly average temperatures) and try to model it using a sine function.

Browse

Browse

Graphing Trigonometric Functions

Learning Objectives

The Anatomy of a Wave

The Rhythm: Period and Phase Shift

The Five-Point Method

Key Takeaways

Quick Check

What's Next?

Graphing Trigonometric Functions

Learning Objectives

The Anatomy of a Wave

The Rhythm: Period and Phase Shift

The Five-Point Method

Key Takeaways

Quick Check

What's Next?