Explain the relationship between the tangent function and the unit circle.

On the unit circle, $\tan(\theta)$ represents the slope of the terminal ray. It's the ratio of the y-coordinate (sine) to the x-coordinate (cosine).

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Explain the relationship between the tangent function and the unit circle.

On the unit circle, $\tan(\theta)$ represents the slope of the terminal ray. It's the ratio of the y-coordinate (sine) to the x-coordinate (cosine).

Why does the tangent function have vertical asymptotes?

Tangent has vertical asymptotes where $\cos(\theta) = 0$ , because $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ . Division by zero is undefined.

Why is the period of the tangent function $\pi$ and not $2\pi$ ?

The slope of the terminal ray on the unit circle repeats every $\pi$ radians (half-rotation), unlike sine and cosine which repeat every $2\pi$ radians.

Explain how the sign of $\tan(\theta)$ changes in different quadrants of the unit circle.

Quadrant I: Positive (both sine and cosine are positive). Quadrant II: Negative (sine is positive, cosine is negative). Quadrant III: Positive (both sine and cosine are negative). Quadrant IV: Negative (sine is negative, cosine is positive).

Describe the behavior of the tangent function near its asymptotes.

As $x$ approaches an asymptote, $\tan(x)$ approaches either positive infinity or negative infinity. The function becomes infinitely steep.

How does a negative 'a' value in $y = a \tan(x)$ affect the graph?

A negative 'a' value reflects the graph of $y = \tan(x)$ over the x-axis.

Explain the effect of changing 'b' in the tangent function.

Changing 'b' in $y = \tan(bx)$ alters the period of the tangent function. A larger 'b' compresses the graph horizontally, decreasing the period, while a smaller 'b' stretches the graph horizontally, increasing the period.

Describe the range of the standard tangent function.

The range of the standard tangent function, $y = \tan(x)$ , is all real numbers, or $(-\infty, \infty)$ .

Explain the concept of phase shift in a tangent function.

Phase shift is the horizontal translation of a tangent function. In the equation $y = a \tan(b(x - c)) + d$ , 'c' represents the phase shift. A positive 'c' shifts the graph to the right, while a negative 'c' shifts it to the left.

What does the 'd' value represent in the equation $y = a \tan(b(x - c)) + d$ ?

The 'd' value represents the vertical shift of the tangent function. A positive 'd' shifts the graph upward, while a negative 'd' shifts it downward.

What is the formula for $\tan(\theta)$ in terms of $\sin(\theta)$ and $\cos(\theta)$ ?

$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$

What is the general equation for a transformed tangent function?

$y = a \tan(b(x - c)) + d$

What is the formula for the period $T$ of a tangent function?

$T = \frac{\pi}{|b|}$

How do you calculate the location of vertical asymptotes for $y = \tan(x)$ ?

$x = \frac{\pi}{2} + k\pi$ , where $k$ is an integer.

How does 'a' affect the tangent function $y = a \tan(x)$ ?

'a' controls the vertical dilation. If 'a' is negative, the function is reflected over the x-axis.

How does 'b' affect the tangent function $y = \tan(bx)$ ?

'b' affects the period of the function. The period is $T = \frac{\pi}{b}$ .

How does 'c' affect the tangent function $y = \tan(x-c)$ ?

'c' shifts the graph horizontally. Positive 'c' shifts right, negative 'c' shifts left.

How does 'd' affect the tangent function $y = \tan(x)+d$ ?

'd' shifts the graph vertically. Positive 'd' shifts up, negative 'd' shifts down.

What is $x$ on the unit circle?

$x = cos(\theta)$

What is $y$ on the unit circle?

$y = sin(\theta)$

What does the steepness of the tangent function's graph indicate?

The steepness indicates the rate of change of the function. Near the asymptotes, the function changes very rapidly.

How can you identify the period of a tangent function from its graph?

The period is the distance between two consecutive vertical asymptotes.

How does the graph of $y = -\tan(x)$ differ from $y = \tan(x)$ ?

The graph of $y = -\tan(x)$ is a reflection of $y = \tan(x)$ over the x-axis.

How can you identify a phase shift from the graph of a tangent function?

Compare the location of the asymptotes to the standard $y = \tan(x)$ graph. A horizontal shift in the asymptotes indicates a phase shift.

What does a vertical shift do to the graph of a tangent function?

It moves the entire graph up or down, changing the y-coordinates of all points on the graph.

How does the graph of $y = a \tan(x)$ change when $|a| > 1$ ?

The graph is vertically stretched, making it steeper compared to the graph of $y = \tan(x)$ .

How does the graph of $y = \tan(bx)$ change when $b > 1$ ?

The graph is horizontally compressed, decreasing the period and bringing the asymptotes closer together.

What does the symmetry of the tangent function about the origin indicate?

It indicates that the tangent function is an odd function, meaning $\tan(-x) = -\tan(x)$ .

If a tangent graph has asymptotes at $x = 0$ and $x = \pi$ , what is its period?

The period is $\pi$ (the distance between the asymptotes).

How can you tell from a graph if a tangent function has been reflected across the x-axis?

The function will decrease from left to right between asymptotes, instead of increasing.