Identifying Key Features

Sketch one period of $y = \tan(x)$ and identify the asymptotes, intercepts, and behavior.

Find the asymptotes: $\cos(x) = 0$ when $x = -\frac{\pi}{2}$ and $x = \frac{\pi}{2}$ = Vertical asymptotes at $x = \pm\frac{\pi}{2}$

Find the x-intercept: $\tan(x) = 0$ when $\sin(x) = 0$, so $x = 0$ = x-intercept at $(0, 0)$

Find key points: $\tan\left(-\frac{\pi}{4}\right) = -1$ and $\tan\left(\frac{\pi}{4}\right) = 1$ = Points: $\left(-\frac{\pi}{4}, -1\right)$ and $\left(\frac{\pi}{4}, 1\right)$

Describe the behavior: As $x \to -\frac{\pi}{2}^+$, $\tan(x) \to -\infty$; as $x \to \frac{\pi}{2}^-$, $\tan(x) \to +\infty$ = Curve rises from $-\infty$ to $+\infty$

Answer: The tangent function has vertical asymptotes at $x = \pm\frac{\pi}{2}$, passes through $(0,0)$, and increases continuously from $-\infty$ to $+\infty$ within each period.

Graphing a Transformed Tangent

Graph $y = 2\tan(x)$ and describe how it differs from $y = \tan(x)$.

Identify the transformation: The coefficient 2 is a vertical stretch = Vertical stretch by factor of 2

Find new key points: When $x = \frac{\pi}{4}$: $y = 2\tan\left(\frac{\pi}{4}\right) = 2 \times 1 = 2$ = Point: $\left(\frac{\pi}{4}, 2\right)$

Check asymptotes: Asymptotes occur where $\cos(x) = 0$ (unchanged) = Same asymptotes at $x = \pm\frac{\pi}{2}$

Check period: No horizontal compression/stretch, so period = $\pi$ = Period unchanged at $\pi$

Answer: The graph of $y = 2\tan(x)$ has the same asymptotes and period as $y = \tan(x)$, but the curve is stretched vertically by a factor of 2, making it steeper.

Finding Period with Horizontal Compression

Find the period and asymptotes of $y = \tan(2x)$.

Recall the period formula: For $y = \tan(bx)$, period = $\frac{\pi}{|b|}$ = Period formula: $\frac{\pi}{|b|}$

Calculate the period: Here $b = 2$, so period = $\frac{\pi}{2}$ = Period = $\frac{\pi}{2}$

Find the asymptotes: Asymptotes where $2x = \frac{\pi}{2} + n\pi$, so $x = \frac{\pi}{4} + \frac{n\pi}{2}$ = Asymptotes at $x = \pm\frac{\pi}{4}, \pm\frac{3\pi}{4}, ...$

Verify with graph behavior: The function completes one full cycle in half the distance = Graph is compressed horizontally

Answer: The period of $y = \tan(2x)$ is $\frac{\pi}{2}$, with asymptotes at $x = \frac{\pi}{4} + \frac{n\pi}{2}$ for integer $n$.

Mistake: Thinking the period of tangent is $2\pi$ like sine and cosine

Why: The tangent function repeats every $\pi$ radians because $\tan(x + \pi) = \tan(x)$ for all $x$ in the domain.

Correct: The period of $y = \tan(x)$ is $\pi$. For $y = \tan(bx)$, the period is $\frac{\pi}{|b|}$.

Mistake: Drawing the curve crossing through the asymptotes

Why: Asymptotes represent values where the function is undefined. The curve approaches but never touches or crosses them.

Correct: Draw vertical dashed lines for asymptotes. The curve approaches $\pm\infty$ as it nears each asymptote.

Mistake: Forgetting that tangent has no maximum or minimum value

Why: Unlike sine and cosine which are bounded between $-1$ and $1$, tangent's range is all real numbers.

Correct: The range of $y = \tan(x)$ is $(-\infty, \infty)$. There is no amplitude for tangent functions.

Road Engineering

Engineers use the tangent function when designing roads with specific gradients and banking angles.

A road rises 5 meters over a horizontal distance of 100 meters. The grade angle $\theta$ satisfies $\tan(\theta) = \frac{5}{100} = 0.05$, giving $\theta \approx 2.86°$.

Aviation Navigation

Pilots use tangent ratios when calculating descent angles and approach paths.

A plane descends from 10,000 feet while traveling 30 miles horizontally. The descent angle $\theta$ has $\tan(\theta) = \frac{10000}{30 \times 5280}$.

Q: Why does tangent have asymptotes but sine and cosine don't?

A: Because $\tan(x) = \frac{\sin(x)}{\cos(x)}$, the function is undefined wherever $\cos(x) = 0$. Division by zero creates vertical asymptotes. Sine and cosine are defined for all real numbers with no division involved.

Q: How do I remember where the asymptotes are?

A: Asymptotes occur where cosine equals zero: at odd multiples of $\frac{\pi}{2}$. That's $\pm\frac{\pi}{2}, \pm\frac{3\pi}{2}, \pm\frac{5\pi}{2}$, etc. Think: halfway between each zero of cosine.

Q: Why is the period $\pi$ instead of $2\pi$?

A: The tangent function repeats after $\pi$ radians because both sine and cosine change sign together after $\pi$, so their ratio stays the same: $\frac{-\sin(x)}{-\cos(x)} = \frac{\sin(x)}{\cos(x)}$.