Trigonometric graphs are mathematical representations of periodic phenomena found in nature and everyday technology, such as the rhythmic electrical impulses of a human heartbeat seen in an electrocardiogram (Normal Sinus Rhythm) or alternating electrical currents. This topic focuses on understanding the fundamental shapes, characteristics, and properties of the basic sine, cosine, and tangent graphs. Students will explore how changing parameters affects amplitude, period, and vertical translation, master the techniques of plotting and sketching transformed curves, and apply graphical methods to determine the roots or number of solutions of trigonometric equations.
Compare the basic curves of \(y = \sin x\), \(y = \cos x\), and \(y = \tan x\), and see how \(a\), \(b\), and \(c\) modify each graph.
Passes through the origin. Maximum \(1\), minimum \(-1\), amplitude \(1\), period \(2\pi\).
Starts at its maximum value at \(x = 0\). Maximum \(1\), minimum \(-1\), amplitude \(1\), period \(2\pi\).
Discontinuous with vertical asymptotes. No amplitude, no maximum or minimum, period \(\pi\).
Understand the basic shapes, transformations, and graphical solution methods for trigonometric functions.
For \(y = \sin x\) and \(y = \cos x\), the wave oscillates smoothly between a highest point (maximum) and a lowest point (minimum) about a central baseline known as the equilibrium line. The vertical height from this baseline to a peak is called the amplitude, which is determined by \(|a|\).
For \(y = a \sin bx + c\) and \(y = a \cos bx + c\): amplitude \(= |a|\), period \(= \dfrac{360^\circ}{b} = \dfrac{2\pi}{b}\), equilibrium line is \(y = c\), maximum \(= c + |a|\), minimum \(= c - |a|\).
The tangent graph behaves fundamentally differently: because tangent is the ratio of sine to cosine, it becomes undefined whenever cosine is zero (at \(90^\circ, 270^\circ\), etc.). This creates vertical barrier lines called asymptotes. Tangent has no crests or troughs, hence it has no maximum, minimum, or amplitude, and its natural cycle repeats every \(180^\circ\) (\(\pi\) radians).
When solving equations graphically, sketching the trigonometric curve alongside a secondary straight line reveals all valid roots as the points where the two curves intersect. The graphical method gives the \(x\)-coordinates of the intersection points as the solutions.
Rearrange the equation into two simultaneous functions: \(y_1 = \text{trigonometric function}\) and \(y_2 = \text{linear or algebraic function}\).
Plot or sketch both \(y_1\) and \(y_2\) on the same set of coordinate axes for the specified domain.
Identify the points of intersection between the two graphs.
The solutions (roots) are the \(x\)-coordinates of the intersection points. The number of solutions equals the number of intersection points.
Period of sine and cosine
Number of cycles in \(0 \le x \le 2\pi\) equals \(b\).
Period of tangent
Tangent has no amplitude, maximum, or minimum.
Answer in your mind, then press "Check Answer".
State the amplitude, period in degrees, and minimum value for the trigonometric function \(y = 4 \cos 3x - 1\).
Find the period of the function \(f(x) = 2 \tan 4x\) in radians and state the equations of the vertical asymptotes in the interval \(0 \le x \le \dfrac{\pi}{2}\).
Determine the number of solutions to the equation \(2\pi \sin 2x - x = 0\) for \(0 \le x \le \pi\).
Modulus graphs just shift upwards: Believing that \(y = |\cos x|\) shifts the graph up by 1 unit.
In reality, any section of the curve below the \(x\)-axis is reflected upwards across the \(x\)-axis, changing the period of the wave to half its original value.
Number of solutions equals number of cycles: Assuming that a function with 2 cycles must have 2 solutions when intersected with a straight line.
The number of solutions strictly depends on the specific points where the line crosses the curve.
Confusing radians and degrees on the \(x\)-axis: Plotting values like \(\pi\) as \(180\) on a Cartesian coordinate plane when intersecting with an algebraic straight line such as \(y = \dfrac{x}{\pi}\).
For linear functions, \(\pi\) must be treated numerically (\(\approx 3.142\)).
Asymptotes being touched: Believing the tangent curve eventually touches the asymptote at extreme heights.
An asymptote is a line that the curve gets arbitrarily close to, but never meets or crosses.
Objective: Identify amplitude, period, asymptotes, transformations, and graphical solution methods for trigonometric functions.
1 What is the amplitude and period of the trigonometric function \(y = -3 \cos 4x + 2\)?
2 Which of the following statements is true regarding the graph of \(y = \tan 2x\) for \(0 \le x \le \pi\)?
3 The period of the trigonometric function \(y = 3 \tan \left(\dfrac{1}{2}x\right)\) is ________ radians.
4 The vertical line which a tangent curve approaches infinitely close to but never touches is known as an ________.
Drag each term to the correct meaning. On a phone, tap a term first, then tap the matching box.
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