Question

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Explanation:

⚡ Using what you learned: Graphing Trigonometric Functions

Step 1: Analyze Graph 1 (Top)

Identify the key features of the first graph:

• It consists of multiple disconnected, repeating curves that increase from left to right.
• The curves have vertical asymptotes.
• One curve passes through the origin \( (0,0) \).
• This is the characteristic shape of the tangent function, \( y = \tan(x) \).
• The period is \( 180^\circ \) (or \( \pi \) radians), with asymptotes at \( x = \pm 90^\circ, \pm 270^\circ \).

Step 2: Analyze Graph 2 (Second from top)

Identify the key features of the second graph:

• It is a continuous wave (sinusoidal).
• At \( x = 0 \), the graph is at its maximum value.
• A wave that starts at its peak at \( x = 0 \) is a cosine function, \( y = \cos(x) \).
• The period is \( 360^\circ \), with a maximum at \( 0^\circ \), a minimum at \( 180^\circ \), and returning to a maximum at \( 360^\circ \).

Step 3: Analyze Graph 3 (Third from top)

Identify the key features of the third graph:

• It is a continuous wave (sinusoidal).
• At \( x = 0 \), the graph passes through the origin \( (0,0) \) and goes upwards to the right.
• A wave that starts at \( 0 \) and goes up is a sine function, \( y = \sin(x) \).
• The period is \( 360^\circ \), with a peak at \( 90^\circ \), crossing zero at \( 180^\circ \), a trough at \( 270^\circ \), and returning to zero at \( 360^\circ \).

Step 4: Analyze Graph 4 (Fourth from top)

Identify the key features of the fourth graph:

• It is a sinusoidal wave with a higher frequency (shorter period).
• At \( x = 0 \), the graph passes through the origin \( (0,0) \) and goes upwards.
• The wave completes one full cycle between \( 0^\circ \) and \( 180^\circ \).
• Since the period is halved (\( 180^\circ \) instead of \( 360^\circ \)), the frequency is doubled: \( y = \sin(2x) \).

Step 5: Analyze Graph 5 (Fifth from top)

Identify the key features of the fifth graph:

• It is a sinusoidal wave with a higher frequency.
• At \( x = 0 \), the graph is at its maximum value.
• The wave completes one full cycle between \( 0^\circ \) and \( 180^\circ \).
• Since it starts at a maximum and has a period of \( 180^\circ \), this is a cosine function with doubled frequency: \( y = \cos(2x) \).

Step 6: Analyze Graph 6 (Bottom)

Identify the key features of the sixth graph:

• It consists of multiple disconnected, repeating curves with vertical asymptotes.
• The curves are packed more closely together than in Graph 1.
• The period is halved to \( 90^\circ \), meaning the function repeats twice as fast.
• This is the tangent function with doubled frequency: \( y = \tan(2x) \).

Answer:

The matching functions for each graph from top to bottom are:

1. First Graph: \( y = \tan(x) \)
2. Second Graph: \( y = \cos(x) \)
3. Third Graph: \( y = \sin(x) \)
4. Fourth Graph: \( y = \sin(2x) \)
5. Fifth Graph: \( y = \cos(2x) \)
6. Sixth Graph: \( y = \tan(2x) \)