Graphing Trigonometric Functions and Comparing Families

Lesson 2 of 2: Trig Functions and Function Comparison

In this lesson:

• Graph and identify amplitude, period, midline
• Use A, B, D to transform sine and cosine graphs
• Compare exponential, logarithmic, and trigonometric families

Learning Objectives for This Lesson

By the end of this lesson, you should be able to:

5. Graph and , identify amplitude, period, and midline
6. Use A, B, D to transform sine/cosine and label max, min, and key features

What Shape Does a Sound Wave Make?

Sound, light, tides, seasons — many real-world phenomena repeat on a cycle.

• Not a J-curve, not a slow-rising curve
• A wave that goes up and down forever

The sine and cosine functions model this repeating, wave behavior.

Parent Sine Function: Five Key Points

One full cycle from to :

Rise to 1, return to 0, fall to −1, return to 0.

Graph of

Amplitude: 1; Period: ; Midline:

Parent Function

One full cycle from to :

Same shape as sine — starts at the maximum instead of zero.

Parameters A, B, and D Explained

• = amplitude — height from midline to peak
• = period — length of one cycle
• = midline — horizontal center; max , min

Amplitude and Period at a Glance

• Amplitude = vertical distance from midline to peak
• Period = horizontal distance for one complete cycle

Quick Check: Amplitude and Period

• What is the amplitude of ?
• What is the period of ?

Think before advancing — one of these is a common error trap.

Worked Example:

Identify parameters:

• , ,

Compute features:

• Amplitude:
• Period:
• Midline:
• Max: ; Min:

Five Key Points for Transformed Sine

One cycle from to (the period):

x-values are quarter-periods; y-values use midline ± amplitude.

Your Turn:

Find:

• , ,
• Amplitude = ?
• Period = ?
• Midline = ?
• Max value = ?, Min value = ?

Complete all five before advancing.

Practice: Find Amplitude, Period, and Midline

For each, find amplitude, period, midline, max, and min:

Work through all three, then advance for answers.

Answers to Practice: Trig Features

1. : amp ; period ; midline ; max , min

2. : amp ; period ; midline ; max , min

3. : amp ; period ; midline ; max , min

Three Function Families: Signature Shapes

Each family has a distinct visual signature.

Comparing All Three Function Families

Real-World Connections for Each Family

• Exponential: compound interest — keeps growing faster over time
• Logarithmic: Richter scale — large physical change maps to small scale change
• Trigonometric: seasonal temperatures — repeating annual cycle

Quick Check: Identify the Family

Which family does each graph belong to?

A. Rises steeply from the left, then flattens; cannot go below zero
B. Oscillates between −4 and 4 with constant period
C. Curves upward on the right; approaches a horizontal line on the left

Identify A, B, and C before the next slide.

Worked: How to Identify a Function Family

To classify an unknown graph, check in order:

1. Does it repeat? → Trigonometric
2. Does it approach a horizontal line on one side? → Exponential
3. Does it approach a vertical line on one side? → Logarithmic

Shape is the first clue; asymptote type confirms it.

Practice: Classify Each Graph by Family

Identify the family and state one key feature:

1. Approaches from below; domain all reals
2. Crosses x-axis at ; defined only for
3. Amplitude 2; midline ; period
4. y-intercept ; decreasing toward

Answers to Graph Classification Practice

1. Exponential — HA at is the signature feature
2. Logarithmic — x-int and domain
3. Trigonometric — amplitude, midline, and period identify it
4. Exponential (decay) — y-int , decreasing toward HA

Key Takeaways from This Lesson

Trigonometric :

• Amplitude ; Period ; Midline
• Five-point method: quarter-period intervals

Comparison: Exponential has HA; Log has VA; Trig repeats.

Watch out: Amplitude is , never . Period , not .

Up Next: Modeling and Inverse Relationships

Coming up in later lessons:

• Modeling periodic phenomena with (HSF.TF.B.5)
• Inverse relationships between exponential and logarithmic functions (HSF.BF.B.5)
• Exponential equations and logarithmic solutions (HSF.LE.A.4)

You've now graphed all three special function families — strong foundation for what's ahead.