The question in the image asks to compare the graphs of two pairs of functions.

Part a:

You're asked to find the difference between the graphs of:

1. $y = x^2$
2. $y = (x - 4)^2$

Answer: The graph of $y = (x - 4)^2$ is a horizontal shift of the graph of $y = x^2$. Specifically, it is shifted 4 units to the right. This happens because the $(x - 4)$ term moves the parabola horizontally by 4 units in the positive $x$-direction.

Part b:

You're asked to find the difference between the graphs of:

1. $y = \frac{1}{3}x^2$
2. $y = 3x^2$

Answer: The graph of $y = \frac{1}{3}x^2$ is a vertical stretch/compression compared to $y = x^2$, and similarly, $y = 3x^2$ is also a vertical stretch. Specifically:

• $y = \frac{1}{3}x^2$ makes the parabola wider, as it compresses the graph vertically.
• $y = 3x^2$ makes the parabola narrower, as it stretches the graph vertically.

Would you like more detailed explanations or visual representations of these transformations?

5 Related Questions:

1. What happens when you subtract a constant inside the parentheses of a quadratic function?
2. How would the graph of $y = (x + 4)^2$ compare to $y = x^2$?
3. What effect does multiplying the quadratic term by a negative coefficient have on the graph?
4. How would you shift a parabola vertically instead of horizontally?
5. Can the vertical stretch factor ever make a parabola appear upside down?

Tip: Horizontal shifts affect only the $x$-coordinate, while vertical transformations (stretch/compression) affect the $y$-coordinate of the graph.