Horizontal and Vertical translation

Horizontal and Vertical Translations of Exponential Functions

• Learning Outcomes
  • Understanding transformations of exponential graphs.
  • Similar behavior of exponential functions to other parent functions concerning shifts, reflections, stretches, and compressions.
  • Ability to graph exponential functions after horizontal and vertical shifts and articulate the associated equations.

Transformations: Overview

• Transformations applied to the function $f(x) = b$ include:
  • Shifts: Translating the graph left/right or up/down.
  • Reflections: Flipping the graph over a specified axis.
  • Stretches/Compressions: Altering the vertical or horizontal stretch/compression of the graph.

Vertical Shifts

• A vertical shift occurs by adding a constant $d$ to the function: $g(x) = b^{x} + d$
  • Example:
  • Parent function:
    $f(x) = 2^{x}$
  • Shift up:
    $g(x) = 2^{x} + 3$
  • Shift down:
    $h(x) = 2^{x} - 3$
• Effects of vertical shifts:
  • The domain remains unchanged: (-, ).
  • The y-intercept shifts accordingly:
  • $g(x)$ shifts to (0, 4) when shifted up, and $h(x)$ shifts to (0, -2) when shifted down.
  • The asymptote moves:
  • For $g(x)$, the asymptote is $y = 3$.
  • For $h(x)$, the asymptote is $y = -3$.
  • The range changes:
  • For $g(x)$:
    • (3, ) .
  • For $h(x)$:
    • (-3, ) .

Horizontal Shifts

• Horizontal shift occurs by adding a constant $c$ to the input of the function:
  $g(x) = b^{x+c}$
• Both left and right shifts are determined by the sign of $c$:
  • Left Shift:
  • Example:
    • $g(x) = 2^{x+3}$
  • Right Shift:
  • Example:
    • $h(x) = 2^{x-3}$
• Effects of horizontal shifts:
  • The domain remains unchanged: (-, ).
  • The asymptote remains unchanged: $y = 0$.
  • The y-intercept adjusts:
  • For the left shift, the new y-intercept of $g(x)$ becomes (0, 8).
  • For the right shift, $h(x)$ has a new y-intercept of (0, 18).

Summary of Functions

• General transformation formula for functions of the form:
  $f(x) = b^{x} + c + d$
• Where:
  • $c$ causes horizontal shifts (negative moves right, positive moves left).
  • $d$ causes vertical shifts (positive moves up, negative moves down).

How to Graph Transformed Functions

1. Start with the horizontal asymptote: $y = d$.
2. Shift the graph of the parent function:
   • Horizontally by $c$ units (left/right based on the sign of $c$).
   • Vertically by $d$ units (up/down based on the sign of $d$).
3. State the domain:
   • (-, ).
4. State the range:
   • (d, )
5. State the horizontal asymptote:
   • $y = d$.

Approximation of Solutions to Exponential Equations

• Graphing can assist in solving exponential equations:
  • Example:
    For the equation $4 = 2^x$, you can plot $f(x) = 2^x$ and see where it meets $y = 4$.

Example Problems

1. To accurately graph and find solutions:
   • Use online graphing calculators to visualize function transformations and intersections.
   • Solve specific equations graphically to find values for $x$ that make them true (such as $4 = 7.85*(1.15)^x - 2.27$).

Important Notes

• Always remember that similar principles apply for transferring across function types, focusing on maintaining the systematic approach to horizontal and vertical translations.