Strand 2 — Geometry & Trigonometry

Trigonometric Graphs: Amplitude, Period, and Range

5th Year · 6th Year (Leaving Cert)

✓By the end of this lesson students will be able to identify the amplitude, period, and range of trigonometric functions of the form y = a sin(bx) + c.
✓By the end of this lesson students will be able to sketch graphs of trigonometric functions based on their amplitude, period, and vertical shift.
✓By the end of this lesson students will be able to determine the equation of a trigonometric graph given its key features.
✓By the end of this lesson students will be able to use both degree and radian measures for the period of trigonometric functions.

Key concepts

General Form of Sine Graph

The general form of a sine function is y = a sin(bx) + c. Each parameter (a, b, c) affects the graph in a specific way: - 'a' determines the amplitude and whether the graph is reflected vertically. - 'b' determines the period (horizontal stretch or compression). - 'c' determines the vertical shift of the graph, which also defines the midline.

y = a sin(bx) + c

Amplitude

The amplitude of a trigonometric graph is the maximum displacement or distance from the midline of the graph to its maximum or minimum point. It is always a positive value. A larger amplitude means a 'taller' graph.

Amplitude = |a|

Period

The period of a trigonometric graph is the length of one complete cycle of the wave. It is the horizontal distance over which the graph repeats itself. For sine and cosine functions, the standard period is 2π radians or 360 degrees. The 'b' value in the general form alters this period.

Period (radians) = 2π / |b| Period (degrees) = 360° / |b|

Range

The range of a function is the set of all possible y-values that the function can take. For a sine graph, the range is determined by its amplitude and its vertical shift (midline). The maximum value is c + |a| and the minimum value is c - |a|.

Range = [c - |a|, c + |a|]

Midline

The midline of a trigonometric graph is the horizontal line that passes exactly halfway between the maximum and minimum values of the function. It represents the vertical shift of the graph. For y = a sin(bx) + c, the midline is y = c.

Midline: y = c

Key facts to remember

1The amplitude is always positive: Amplitude = |a|.
2The period for y = a sin(bx) + c is 2π/|b| (radians) or 360°/|b| (degrees).
3The midline of the graph is the horizontal line y = c.
4The range of the graph is [c - |a|, c + |a|].
5A negative 'a' value indicates a vertical reflection across the midline.
6The 'b' value affects the horizontal compression or stretch, determining the period.
7The 'c' value shifts the entire graph vertically.

Worked examples

Example 1

For the function y = 3 sin(2x), determine its amplitude, period (in both degrees and radians), and range.

IIdentify the values of a, b, and c from the general form y = a sin(bx) + c.

IIIn y = 3 sin(2x), we have a = 3, b = 2, and c = 0 (since there is no vertical shift).

IIICalculate the amplitude: Amplitude = |a| = |3| = 3.

IVCalculate the period in radians: Period = 2π / |b| = 2π / |2| = π radians.

VCalculate the period in degrees: Period = 360° / |b| = 360° / |2| = 180°.

VICalculate the range: Range = [c - |a|, c + |a|] = [0 - 3, 0 + 3] = [-3, 3].

Answer

Amplitude = 3, Period = π radians (or 180°), Range = [-3, 3].

When c = 0, the midline is the x-axis (y=0).

Example 2

Find the amplitude, period, and range of the function y = -2 sin(πx) + 1. Sketch one complete cycle of the graph.

IIdentify the values of a, b, and c: a = -2, b = π, c = 1.

IICalculate the amplitude: Amplitude = |a| = |-2| = 2.

IIICalculate the period: Period = 2π / |b| = 2π / |π| = 2 radians.

IVCalculate the range: Range = [c - |a|, c + |a|] = [1 - 2, 1 + 2] = [-1, 3].

VDetermine the midline: y = c = 1.

VIIdentify key points for sketching: The graph starts at the midline (y=1). Since a is negative, it will go down first instead of up.

VIIStarting point (x=0): y = -2 sin(0) + 1 = 1. So, (0, 1).

VIIIQuarter period (x = 2/4 = 0.5): The graph reaches its minimum. y = -2 sin(π/2) + 1 = -2(1) + 1 = -1. So, (0.5, -1).

9Half period (x = 2/2 = 1): The graph returns to the midline. y = -2 sin(π) + 1 = -2(0) + 1 = 1. So, (1, 1).

10Three-quarter period (x = 3*2/4 = 1.5): The graph reaches its maximum. y = -2 sin(3π/2) + 1 = -2(-1) + 1 = 3. So, (1.5, 3).

11End of period (x = 2): The graph returns to the midline. y = -2 sin(2π) + 1 = -2(0) + 1 = 1. So, (2, 1).

12Sketch the points and draw a smooth curve through them.

Answer

Amplitude = 2, Period = 2, Range = [-1, 3]. (Sketch would show a sine wave starting at (0,1), going down to (0.5,-1), up to (1,1), further up to (1.5,3), and back down to (2,1).)

A negative 'a' value means the graph is reflected across its midline.

Example 3

A sine graph has a maximum value of 5 and a minimum value of -1. It completes one cycle in 720°. Find the equation of the graph in the form y = a sin(bx) + c.

ICalculate the amplitude: Amplitude = (Max Value - Min Value) / 2 = (5 - (-1)) / 2 = 6 / 2 = 3. So, |a| = 3. We assume a > 0 unless otherwise specified, so a = 3.

IICalculate the midline (c): Midline = (Max Value + Min Value) / 2 = (5 + (-1)) / 2 = 4 / 2 = 2. So, c = 2.

IIIUse the given period to find 'b': Period = 360° / |b|. We are given Period = 720°.

IV720° = 360° / |b| => |b| = 360° / 720° = 1/2. We assume b > 0, so b = 1/2.

VSubstitute the values of a, b, and c into the general form y = a sin(bx) + c.

VIy = 3 sin((1/2)x) + 2.

Answer

The equation of the graph is y = 3 sin(x/2) + 2.

Always check if the graph starts at the midline and goes up or down to determine the sign of 'a'.

Common mistakes

✗Confusing the 'b' value with the period itself; remember to use the formula 2π/|b| or 360°/|b|.
✗Forgetting to take the absolute value of 'a' when calculating amplitude, or 'b' when calculating period.
✗Incorrectly calculating the range, especially when 'c' is negative or 'a' is negative.
✗Assuming 'a' is always positive; a negative 'a' value means the graph starts by going down from the midline (for sine).
✗Mixing up radians and degrees when calculating the period; always check the units required or implied by the problem.

Exam tips

★Always clearly state the values of a, b, and c before calculating amplitude, period, and range.
★When sketching, identify the midline, maximum, minimum, and the start/end points of one cycle. Use quarter-period intervals for key points.
★Pay close attention to whether the question specifies radians or degrees for the period.
★If finding the equation from a graph, first determine the midline and amplitude, then the period, and finally consider any reflections.

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