Ket f(x)=2x². Explain the effect of the transformations that is g(x).                                                                                                                     a) g(x) = 3f(x)            b) g(x)=f(x+3)              c)g(x)= f(x)-3                 dg(x)= f(1/2x)             

Let's analyze each transformation applied to the function \( f(x) = 2x^2 \) and the resulting effect on the graph of \( f(x) \):

### a) \( g(x) = 3f(x) \)
This transformation multiplies the output of the original function by 3.
- **Transformation**: Vertical stretch.
- **Effect**: The graph of \( f(x) \) is stretched vertically by a factor of 3. This means that for every \( x \), the corresponding \( y \)-values are three times larger than they were in \( f(x) \). The parabola becomes narrower.

### b) \( g(x) = f(x+3) \)
This is a horizontal shift.
- **Transformation**: Horizontal translation.
- **Effect**: The graph of \( f(x) \) is shifted to the left by 3 units. Each point on the graph moves 3 units in the negative \( x \)-direction.

### c) \( g(x) = f(x) - 3 \)
This transformation subtracts 3 from the output.
- **Transformation**: Vertical translation.
- **Effect**: The graph of \( f(x) \) is shifted downward by 3 units. Every point on the graph moves 3 units down along the \( y \)-axis.

### d) \( g(x) = f\left(\frac{1}{2}x\right) \)
This transformation involves scaling the input of the function.
- **Transformation**: Horizontal stretch.
- **Effect**: The graph of \( f(x) \) is stretched horizontally by a factor of 2. The \( x \)-values are scaled, making the parabola wider. For each \( x \)-value, the function now reaches the same height over a larger range.

### Summary of effects:
- \( g(x) = 3f(x) \): **Vertical stretch** by a factor of 3.
- \( g(x) = f(x+3) \): **Horizontal shift** 3 units to the left.
- \( g(x) = f(x) - 3 \): **Vertical shift** 3 units down.
- \( g(x) = f\left(\frac{1}{2}x\right) \): **Horizontal stretch** by a factor of 2.

Would you like more details or have any questions? 

Here are 5 related questions to think about:
1. What would happen if the transformation was \( g(x) = \frac{1}{2}f(x) \)?
2. How does the transformation \( g(x) = f(x - 4) \) affect the graph?
3. What is the effect of the transformation \( g(x) = f(-x) \)?
4. If \( g(x) = f(2x) \), what would be the change in the graph?
5. How would the transformation \( g(x) = f(x) + 5 \) impact the function?

**Tip**: Always focus on how transformations affect either the \( x \)-values (horizontal) or the \( y \)-values (vertical) when analyzing function changes!

Ket f(x)=2x². Explain the effect of the transformations that is g(x).
 a) g(x) = 3f(x) 
 b) g(x)=f(x+3) 
 c)g(x)= f(x)-3 
 d)g(x)= f(1/2x)

Explore how different transformations such as vertical stretches, horizontal shifts, and scaling affect the quadratic function f(x) = 2x². Learn about function transformations like g(x) = 3f(x), g(x) = f(x+3), and more. Suitable for Grades 9-11 students.

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