Transformations of Exponential Graphs

Types of Transformations

Horizontal and Vertical Shifts

• Formula: Graph of $f(x + h) + k$ is $f(x)$ shifted left by $h$ and up by $k$.
  • Negative $h$: Shift right.
  • Positive $h$: Shift left.
  • Positive $k$: Shift up.
  • Negative $k$: Shift down.
• Example: $G(x) = b^{x+h} + k$
  • Horizontal asymptote at $y = k$.
  • Convenient point when exponent is 0: $(-h, k+1)$.
  • Another point when exponent is 1: $(-h+1, b+k)$.

Example of Shift

• Function: $f(x) = 2^{x+3} - 1$
  • $h = 3$, shift left 3 units.
  • $k = -1$, shift down 1 unit.
  • Plot points:
    • $(-3, 0)$ when $x = -3$.
    • $(-2, 1)$ when $x = -2$.
    • $y$-intercept $(0, 7)$ when $x = 0$.
  • Horizontal asymptote: $y = -1$.

Vertical Stretches

• Formula: Graph of $a \times f(x)$ is $f(x)$ stretched vertically by $a$.
  • $a > 1$: True stretching (upward).
  • $0 < a < 1$: Compression.
• Example: $G(x) = a \times b^x$
  • Horizontal asymptote remains $y = 0$.
  • Points: $(0, a)$ and $(1, a \times b)$.

Example of Vertical Stretch

• Function: $f(x) = 3^x$
  • Stretch by $a = 2$: Points are $(0, 2)$ and $(1, 6)$.
  • Stretch by $a = 3$: Points are $(0, 3)$.
  • Adding 1 shifts horizontal asymptote to $y = 1$.

Reflections

• Reflection over Y-axis: $f(-x)$ reflects over the y-axis.
• Reflection over X-axis: $-f(x)$ reflects over the x-axis.

Example of Reflection

• Function: $f(x) = -2 \times 3^{-x}$
  • Reflecting over both axes.
  • Convenient points:
    • $(0, -2)$ when $x = 0$.
    • $(-1, -6)$ when $x = -1$.
  • Horizontal asymptote: y = 0.

Final Note

• Understanding these transformations helps in sketching and predicting the behavior of exponential graphs in various transformations.