The Main Idea

The graphs of [latex]y=\tan x[/latex] and [latex]y=\cot x[/latex] look different from sine and cosine because they have vertical asymptotes and repeat every [latex]\pi[/latex] units (instead of [latex]2\pi[/latex]). Variations of these graphs come from changing the parameters in the general forms:

• [latex]y=a\tan(bx-c)+d[/latex]

• [latex]y=a\cot(bx-c)+d[/latex]

Each parameter changes the graph in predictable ways:

• [latex]a[/latex] stretches or reflects the graph vertically.

• [latex]b[/latex] changes the period (the length of one cycle).

• [latex]c[/latex] shifts the graph left or right (phase shift).

• [latex]d[/latex] shifts the graph up or down (vertical shift).

Quick Tips: Graphing Variations of Tangent and Cotangent

1. Base Graph Features

   • [latex]y=\tan x[/latex]: period [latex]\pi[/latex], asymptotes at [latex]x=\dfrac{\pi}{2}+k\pi[/latex].

   • [latex]y=\cot x[/latex]: period [latex]\pi[/latex], asymptotes at [latex]x=k\pi[/latex].

2. Period Changes

   • Formula: [latex]\text{Period}=\dfrac{\pi}{b}[/latex].

   • Larger [latex]b[/latex] = compressed, smaller [latex]b[/latex] = stretched.

3. Phase Shift [latex]\dfrac{c}{b}[/latex]

   • Moves the graph left or right.

   • Asymptotes shift accordingly.

4. Vertical Stretch/Reflection

   • [latex]a[/latex] changes steepness of the graph.

   • Negative [latex]a[/latex] reflects across the x-axis.

5. Vertical Shift [latex]d[/latex]

   • Moves the midline up or down.

   • Asymptotes remain vertical.

6. Graphing Strategy

   • Step 1: Find the period using [latex]\dfrac{\pi}{b}[/latex].

   • Step 2: Locate the asymptotes.

   • Step 3: Apply phase and vertical shifts.

   • Step 4: Plot key points ([latex]\pm 1[/latex] for tangent; reciprocal behavior for cotangent).

   • Step 5: Sketch the curve between asymptotes.

The Main Idea

To write a function formula from a tangent or cotangent graph, we analyze its key features: the period, the phase shift (location of asymptotes or intercepts), any vertical shift, and the steepness of the curve. Tangent and cotangent graphs both repeat every [latex]\pi[/latex], but parameters in the general forms

• • • • • • [latex]y=a\tan(bx-c)+d[/latex]

          • [latex]y=a\cot(bx-c)+d[/latex]

control how the graph is stretched, shifted, or reflected. By identifying these features from the graph, we can reconstruct the exact equation.

Quick Tips: Building Tangent or Cotangent Formulas

1. Determine the Period

   • For tangent and cotangent, [latex]\text{Period}=\dfrac{\pi}{b}[/latex].

   • Measure the distance between consecutive asymptotes (or repeating points) to find [latex]b[/latex].

2. Locate Phase Shift

   • Tangent asymptotes: [latex]x=\dfrac{c}{b}+\dfrac{\pi}{2b}+k\dfrac{\pi}{b}[/latex].

   • Cotangent asymptotes: [latex]x=\dfrac{c}{b}+k\dfrac{\pi}{b}[/latex].

   • Identify where the central asymptote (for tangent) or intercept (for cotangent) has shifted.

3. Find Vertical Shift [latex]d[/latex]

   • Midline of the graph is [latex]y=d[/latex].

   • Check if the curve has been moved up or down.

4. Determine [latex]a[/latex] (Stretch/Reflection)

   • [latex]a[/latex] changes the steepness.

   • Negative [latex]a[/latex] flips the graph across the midline.

5. Choose Tangent or Cotangent Form

   • Tangent passes through the origin (before shifts) and increases left to right.

   • Cotangent decreases left to right, starting with an asymptote at the origin.

6. Write the Equation

   • Plug amplitude [latex]a[/latex], period factor [latex]b[/latex], phase shift [latex]c[/latex], and vertical shift [latex]d[/latex] into [latex]y=a\tan(bx-c)+d[/latex] or [latex]y=a\cot(bx-c)+d[/latex].

The Main Idea

The graphs of [latex]y=\sec x[/latex] and [latex]y=\csc x[/latex] are built from cosine and sine, since [latex]\sec x=\dfrac{1}{\cos x}[/latex] and [latex]\csc x=\dfrac{1}{\sin x}[/latex]. They feature repeating U-shaped and inverted U-shaped branches with vertical asymptotes where sine or cosine equals zero. Variations of these graphs are created by changing the parameters in the general forms:

• • • • • • [latex]y=a\sec(bx-c)+d[/latex]

          • [latex]y=a\csc(bx-c)+d[/latex]

Each parameter controls how the graph is stretched, shifted, or reflected.

Quick Tips: Graphing Variations of Secant and Cosecant

1. Base Graph Features

   • [latex]y=\sec x[/latex]: period [latex]2\pi[/latex], asymptotes at [latex]x=\dfrac{\pi}{2}+k\pi[/latex].

   • [latex]y=\csc x[/latex]: period [latex]2\pi[/latex], asymptotes at [latex]x=k\pi[/latex].

2. Period Changes

   • Formula: [latex]\text{Period}=\dfrac{2\pi}{b}[/latex].

   • Adjusts how wide each repeating cycle is.

3. Phase Shift [latex]\dfrac{c}{b}[/latex]

   • Moves the graph left or right.

   • Asymptotes and branches shift accordingly.

4. Vertical Stretch/Reflection [latex]a[/latex]

   • [latex]a[/latex] changes the distance from the midline to the minimum/maximum points of each branch.

   • Negative [latex]a[/latex] reflects the branches across the midline.

5. Vertical Shift [latex]d[/latex]

   • Moves the midline up or down, shifting the entire graph.

   • Asymptotes stay vertical, but branch positions adjust.

6. Graphing Strategy

   • Step 1: Start with sine or cosine as a guide.

   • Step 2: Identify asymptotes where sine or cosine = 0.

   • Step 3: Plot key points at maximum/minimum distances from the midline.

   • Step 4: Sketch U-shaped and inverted U-shaped branches between asymptotes.

The Main Idea

To determine a function formula from a secant or cosecant graph, we use the fact that these graphs are built as reciprocals of cosine and sine. The graph’s midline, vertical shift, period, phase shift, and stretch/reflection can all be read directly from its repeating U-shaped or inverted U-shaped branches. Once these features are identified, they are plugged into the general forms:

• • • • • • [latex]y=a\sec(bx-c)+d[/latex]

          • [latex]y=a\csc(bx-c)+d[/latex]

Recognizing the asymptotes and midline first helps anchor the equation.

Quick Tips: Building Secant or Cosecant Formulas

1. Determine the Period

   • Formula: [latex]\text{Period}=\dfrac{2\pi}{b}[/latex].

   • Measure the distance between repeating branches or asymptotes.

2. Locate the Phase Shift

   • Secant asymptotes: align with where cosine = 0, i.e. [latex]x=\dfrac{\pi}{2}+k\pi[/latex].

   • Cosecant asymptotes: align with where sine = 0, i.e. [latex]x=k\pi[/latex].

   • Compare the shifted asymptotes to find [latex]\dfrac{c}{b}[/latex].

3. Find the Vertical Shift [latex]d[/latex]

   • The midline of the graph is [latex]y=d[/latex].

   • This is halfway between a maximum and minimum.

4. Identify the Stretch/Reflection [latex]a[/latex]

   • Distance from the midline to the “top” or “bottom” of a branch = [latex]|a|[/latex].

   • If the branch opens downward where it normally opens upward, [latex]a[/latex] is negative.

5. Choose Secant vs. Cosecant

   • If the branches line up with cosine (max/min at [latex]x=0[/latex]), use secant.

   • If the branches line up with sine (crossing at [latex]x=0[/latex]), use cosecant.

6. Write the Equation

   • Substitute values of [latex]a[/latex], [latex]b[/latex], [latex]c[/latex], and [latex]d[/latex] into the formula.

   • Example: A secant graph with midline [latex]y=1[/latex], amplitude [latex]2[/latex], period [latex]\pi[/latex], and shift right [latex]\dfrac{\pi}{4}[/latex] would be
     [latex]y=2\sec!\left(2x-\dfrac{\pi}{2}\right)+1[/latex].