Examples of function transformations solved

📚 Quick Study Guide

• 📈 Vertical Shifts: Adding a constant *c* to a function shifts it vertically. $f(x) + c$ shifts the graph up by *c* units, and $f(x) - c$ shifts it down by *c* units.
• ↔️ Horizontal Shifts: Replacing *x* with $(x - c)$ shifts the graph horizontally. $f(x - c)$ shifts the graph right by *c* units, and $f(x + c)$ shifts it left by *c* units.
• Stretch: Vertical Stretch: Multiplying a function by a constant *a* stretches it vertically. $a \cdot f(x)$ stretches the graph vertically if $a > 1$ and compresses it if $0 < a < 1$.
• 🗜️ Horizontal Stretch: Replacing *x* with $(ax)$ compresses or stretches the graph horizontally. $f(ax)$ compresses the graph horizontally if $a > 1$ and stretches it if $0 < a < 1$.
• Flip: Reflection over x-axis: Multiplying the function by -1, $-f(x)$, flips the graph over the x-axis.
• 🪞 Reflection over y-axis: Replacing x with -x, $f(-x)$, flips the graph over the y-axis.

Practice Quiz

1. What transformation occurs when $f(x)$ is changed to $f(x) + 3$?
   1. Vertical shift up by 3 units
   2. Vertical shift down by 3 units
   3. Horizontal shift right by 3 units
   4. Horizontal shift left by 3 units
2. What transformation occurs when $f(x)$ is changed to $f(x - 2)$?
   1. Horizontal shift right by 2 units
   2. Horizontal shift left by 2 units
   3. Vertical shift up by 2 units
   4. Vertical shift down by 2 units
3. What transformation occurs when $f(x)$ is changed to $2f(x)$?
   1. Vertical stretch by a factor of 2
   2. Vertical compression by a factor of 2
   3. Horizontal stretch by a factor of 2
   4. Horizontal compression by a factor of 2
4. What transformation occurs when $f(x)$ is changed to $f(3x)$?
   1. Horizontal compression by a factor of 3
   2. Horizontal stretch by a factor of 3
   3. Vertical stretch by a factor of 3
   4. Vertical compression by a factor of 3
5. What transformation occurs when $f(x)$ is changed to $-f(x)$?
   1. Reflection over the x-axis
   2. Reflection over the y-axis
   3. Vertical shift down by 1 unit
   4. Horizontal shift right by 1 unit
6. What transformation occurs when $f(x)$ is changed to $f(-x)$?
   1. Reflection over the y-axis
   2. Reflection over the x-axis
   3. Vertical shift up by 1 unit
   4. Horizontal shift right by 1 unit
7. What transformation occurs when $f(x)$ is changed to $f(x+1) - 2$?
   1. Horizontal shift left by 1 unit, vertical shift down by 2 units
   2. Horizontal shift right by 1 unit, vertical shift up by 2 units
   3. Horizontal shift right by 1 unit, vertical shift down by 2 units
   4. Horizontal shift left by 1 unit, vertical shift up by 2 units