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Understanding and Graphing Secant Functions

Apr 22, 2025

Graphing Secant Functions

Key Components of Secant and Cosecant Functions

  • Amplitude: Not applicable as secant and cosecant functions do not have amplitude, but
    • Absolute value of "A" vertically stretches or compresses the function.
  • Period:
    • Formula: ( \frac{2\pi}{B} )
    • Essential for determining the repeat interval.
  • Vertical Shift (C):
    • Shifts the graph up if C is positive.
    • Shifts down if C is negative.
  • Horizontal Shift (D):
    • Shifts right if D > 0 (( x - D )).
    • Shifts left if D is negative (( x + D )).

Example: Graphing a Secant Function

  • Given Function: ( y = 2 \times \sec(4(x-1)) )
  • Factor the Expression: ( 4x - 4 ) becomes ( 4(x - 1) )
  • Period Calculation:
    • ( \frac{2\pi}{4} = \frac{\pi}{2} ) radians (Approx. 1.57)
  • Horizontal Shift:
    • ( x - 1 ) shifts the graph right by 1 unit.

Graphing Process

  • Secant and Cosine Relationship:
    • Secant and cosine functions are reciprocals.
    • Often helpful to graph cosine to understand the secant.
    • Vertical Asymptotes in secant where cosine is zero.
  • Graphing the Cosine Function:
    • Begin with the basic cosine function from 0 to ( 2\pi ).
    • Adjust for the transformation: ( y = 2 \times \cos(4(x - 1)) ).
    • Plot Key Points:
      • Period is divided into 4 equal parts to mark cosine values.
      • Shift right by 1 unit.
      • Plot points: start at max, zero, min, zero, max.
      • Multiply by coefficient of 2 for height.
  • Repeat the Graph:
    • Repeat cosine function for additional periods left and right.
    • Divide intervals (1.57 units) into four parts.
    • Follow same pattern for max, zero, min, zero, max.

Plotting the Secant Function

  • Use Cosine Graph for Reference:
    • Vertical Asymptotes where cosine = 0.
    • Secant shares max/min values with cosine.
    • Sketch secant curves passing through max/min points, approaching asymptotes.

Conclusion

  • Graphing cosine first aids in sketching secant.
  • Understanding transformations crucial for accurate graphing.
  • Further extensions can be plotted as needed.

The lecture provides a detailed guide on graphing secant functions by leveraging the cosine function as a reference point, emphasizing transformations, period, and shifts.

Full transcript