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!