To tackle problems involving transformations of functions, let's first understand what transformations mean in this context. Transformations generally include shifts (translations), reflections, stretches, or compressions of the graph of a function $f(x)$.

Here’s a step-by-step guide to writing a function $g(x)$ that represents the indicated transformation of the graph of $f(x)$:

Common Transformations:

1. Vertical Shift:

   • Upward by $c$: $g(x) = f(x) + c$
   • Downward by $c$: $g(x) = f(x) - c$

2. Horizontal Shift:

   • Right by $c$: $g(x) = f(x - c)$
   • Left by $c$: $g(x) = f(x + c)$

3. Vertical Stretch/Compression:

   • Stretch by a factor of $a$: $g(x) = a \cdot f(x)$ (where $a > 1$)
   • Compression by a factor of $a$: $g(x) = a \cdot f(x)$ (where $0 < a < 1$)

4. Horizontal Stretch/Compression:

   • Stretch by a factor of $b$: $g(x) = f\left(\frac{x}{b}\right)$ (where $b > 1$)
   • Compression by a factor of $b$: $g(x) = f\left(\frac{x}{b}\right)$ (where $0 < b < 1$)

5. Reflection:

   • Across the x-axis: $g(x) = -f(x)$
   • Across the y-axis: $g(x) = f(-x)$

Example Problem:

If you’re asked to find $g(x)$ for a specific transformation, let’s consider a few scenarios:

1. Shift $f(x)$ upward by 3 units: $g(x) = f(x) + 3$

2. Shift $f(x)$ to the right by 2 units: $g(x) = f(x - 2)$

3. Reflect $f(x)$ across the x-axis and then shift it downward by 4 units: $g(x) = -f(x) - 4$

4. Stretch $f(x)$ vertically by a factor of 2: $g(x) = 2f(x)$

5. Compress $f(x)$ horizontally by a factor of 1/3: $g(x) = f(3x)$

Verification:

To verify these transformations using a graphing calculator:

1. Enter the original function $f(x)$ into the calculator.
2. Enter the transformed function $g(x)$.
3. Compare the graphs of $f(x)$ and $g(x)$ to ensure the transformation was applied correctly.

Would you like to see details on a specific example? You can upload an image or describe a particular function $f(x)$ and transformation you're working on.

Follow-Up Questions:

1. What transformations would reflect the graph across the y-axis and then shift it 2 units left?
2. How does a vertical stretch by a factor of 3 affect the graph of $f(x)$?
3. What happens to the graph of $f(x)$ when it’s reflected across both axes?
4. If $f(x)$ is shifted down by 5 units and then compressed horizontally by a factor of 2, what’s the resulting $g(x)$?
5. How does a shift to the right by 4 units change the equation of $f(x)$?

Tip:

When shifting graphs horizontally, remember that positive values inside the function $f(x - c)$ shift it to the right, while negative values shift it to the left.