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Types of Functions

Types of Functions. Type 1: Constant Function. f(x) = c Example: f(x) = 1. Type 2: Power Function. f(x) = x a. If a is a (+) integer. f(x) = x n where n = 1,2,3,4,5….. -Shape depends on if n is even or odd

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Types of Functions

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  1. Types of Functions

  2. Type 1: Constant Function f(x) = c Example: f(x) = 1

  3. Type 2: Power Function f(x) = xa

  4. If a is a (+) integer f(x) = xn where n = 1,2,3,4,5….. -Shape depends on if n is even or odd -As n increases the graph becomes flatter near 0 and steeper where x ≥ 1

  5. If a is -1 f(x) = x-1 = 1/x Hyperbola

  6. If a = 1/n Root Function f(x) = x1/n = n√(x)

  7. Polynomial f(x) = axn + bxn – 1 + cx n – 2 …….. Degree (n) – highest exponent value 1st Degree: f(x) = ax + b

  8. 2nd Degree: Quadratic: f(x) = ax2+ bx + c Parabola

  9. Higher Degrees

  10. Type 3: Algebraic Functions Can be constructed using algebraic operations (add, subtract, multiplication, division, square root) f(x) = √(x2 + 1) f(x) = x4 – 16x2 + (x-2)3√(x) x + √(x) Shapes vary

  11. Type 4: Trigonometric Functions

  12. Tan(x) = sin(x)/ cos(x)

  13. Type 5: Exponential Functions f(x) = ax

  14. Type 6: Log Function f(x) = logax Inverse exponential

  15. Related Functions • By applying certain transformations to graphs of given functions, we can obtain the graphs of related functions

  16. Translations - Shifts • Vertical shifts • y = f(x) + c shifts c units up • y = f(x) – c shifts c units down

  17. Horizontal shifts y = f(x – c) shifts right c units y = f(x + c) shifts left c units

  18. Stretching and Compressing y = cf(x) stretched vertically by a factor of c y = 1/c f(x) compressed vertically by a factor of c y = f(cx) compressed horizontally by a factor of c y = f(x/c) strectched horizontally by a factor of c

  19. Reflecting y = -f(x) graph reflects about the x-axis y = f(-x) graph reflects about the y-axis

  20. Examples Given y = √(x), sketch a) y = √(x) - 2 b) y = √(x - 2) c) y = - √(x) d) y = 2√(x) e) y = √(-x)

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