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Thursday, November 20 th

Warm Up . Thursday, November 20 th. Write a new equation g(x) compared to the parent function Shift down 4 Shift left 3. Definition of Exponential Functions.

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Thursday, November 20 th

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  1. Warm Up Thursday, November 20th Write a new equation g(x) compared to the parent function Shift down 4 Shift left 3

  2. Definition of Exponential Functions • The exponential function f(x) with a base b is defined by f(x) = bx where b is a positive constant other than 1 (b > 0, and b ≠ 1) and x is any real number. • So, f(x) = 2x, looks like:

  3. Graphing Exponential Functions • Four exponential functions have been graphed. • Compare the graphs of functions where b > 1 to those where b < 1

  4. So, when b > 1, f(x) has a graph that goes up to the right and is an increasing function. EXPONENTIAL GROWTH • When 0 < b < 1, f(x) has a graph that goes down to the right and is a decreasing function. EXPONENTIAL DECAY

  5. Characteristics • The domain of f(x) = bx consists of all real numbers (-, ). • The range of f(x) = bx consists of all positive real numbers (0, ). • The graphs of all exponential functions pass through the point (0,1). This is because f(o) = b0 = 1 (bo). • The graph of f(x) = bx approaches but does not cross the x-axis. The x-axis is a horizontal asymptote. • f(x) = bx is one-to-one and has an inverse that is a function.

  6. Asymptote A line that a graph gets closer and closer to, but never crosses or touches

  7. Transformations • Vertical translation (shift) f(x) = bx + c • Shifts the graph up if c > 0 • Shifts the graph down if c < 0 Remember: the y-intercept in the parent function is 1!

  8. Transformations • Horizontal translation (shift): g(x)=bx+c • Shifts the graph to the left if c > 0 • Shifts the graph to the right if c < 0

  9. Exponential Functions—Stretch or Compressions

  10. Vertical Stretch & Compression Vertical stretching or compression, f(x)=cbx: • Stretches the graph if c > 1 • Compresses the graph if 0 < c < 1

  11. Remember Vertical Stretch= closer to y Vertical Compression=farther away from y

  12. Horizontal Stretch & Compression Horizontal stretching or shrinking, f(x)=bcx: • Compresses the graph if c > 1 • Stretches the graph if 0 < c < 1

  13. Remember Horizontal Stretch= closer to x Horizontal Compression=farther away from x

  14. Reflections g(x) = -bxreflects the graph about the x-axis. g(x) = b-x reflects the graph about the y-axis.

  15. You Try! • Graph the function f(x) = 2(x-3) +2 • Where is the horizontal asymptote? y = 2 *The shift

  16. Graph the function f(x) = 4(x+5) - 3 • Where is the horizontal asymptote? y = - 3 *The shift

  17. Transformations of graphs Topic: What is it? Shifting, stretching, shrinking, and reflecting of graphs Vertical or Horizontal shift Vertical Stretch or Compression Types: Reflection Add outside Multiply by Fraction (less than 1) Multiply by negative (-) MOVES _______________ Causes the graph to _____________________ _____________________ Causes the graph to ____________________ Subtract outside MOVES _______________ Examples Multiply by integer Add inside Causes the graph to ____________________ MOVES _____________ Subtract inside MOVES _____________

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