Transformation problems on a graph with answers 

Here are some typical transformation problems on a graph, along with their answers:

### 1. **Vertical Translation**
   Consider the function \( f(x) = x^2 \). Apply a vertical translation of 3 units upwards.
   
   **Solution:**
   The new function after the transformation is \( g(x) = x^2 + 3 \). This translates every point of the graph of \( f(x) \) upwards by 3 units.

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### 2. **Horizontal Translation**
   Consider the function \( f(x) = \sqrt{x} \). Apply a horizontal translation 2 units to the right.
   
   **Solution:**
   The new function is \( g(x) = \sqrt{x - 2} \). This shifts the graph of \( f(x) \) 2 units to the right.

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### 3. **Reflection across the x-axis**
   Consider the function \( f(x) = \sin(x) \). Reflect the graph across the x-axis.
   
   **Solution:**
   The new function is \( g(x) = -\sin(x) \). This flips the graph of \( f(x) \) over the x-axis, making all positive values negative and vice versa.

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### 4. **Vertical Stretch**
   Consider the function \( f(x) = x^3 \). Apply a vertical stretch by a factor of 2.
   
   **Solution:**
   The new function is \( g(x) = 2x^3 \). This makes the graph steeper, as all y-values are doubled.

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### 5. **Horizontal Compression**
   Consider the function \( f(x) = \cos(x) \). Apply a horizontal compression by a factor of 1/2.
   
   **Solution:**
   The new function is \( g(x) = \cos(2x) \). The graph is compressed horizontally, meaning the graph cycles through its period twice as fast as before.

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Would you like more details or have any questions on these? Here are some related questions:

1. How do you determine the direction of a horizontal translation?
2. What happens to the graph of a function when it undergoes a vertical stretch?
3. What is the effect of reflecting a function across the y-axis instead of the x-axis?
4. Can you explain how a horizontal stretch differs from a vertical stretch?
5. How does adding or subtracting a constant inside a function's argument affect its graph?

**Tip:** Always pay attention to whether the transformation is applied inside or outside the function to determine if it affects the x-values (horizontal changes) or y-values (vertical changes).

Transformation problems on a graph with answers

Explore common transformation problems on a graph, including vertical and horizontal translations, reflections, and stretching or compressing functions. Learn key concepts such as shifting graphs, flipping functions, and altering their steepness. Suitable for students in Grades 9-12.

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