The two equations you provided are quadratic functions, and both represent parabolas. Let's analyze and describe how to graph each one.

1. Equation 1: $y = (x - 3)^2$

• This is a standard form of a parabola, $y = (x - h)^2 + k$, where:
  • $h = 3$ (shifts the graph horizontally to the right by 3 units),
  • $k = 0$ (no vertical shift).
• The vertex of the parabola is at $(3, 0)$, and since the coefficient of $(x - 3)^2$ is positive, the parabola opens upwards.
• The axis of symmetry is $x = 3$.
• This graph is symmetrical around $x = 3$, and its shape is the same as the standard parabola $y = x^2$, just shifted.

2. Equation 2: $y = (x + 2)^2 + 4$

• This is also in the form $y = (x - h)^2 + k$, where:
  • $h = -2$ (shifts the graph horizontally to the left by 2 units),
  • $k = 4$ (shifts the graph vertically up by 4 units).
• The vertex of this parabola is at $(-2, 4)$, and it also opens upwards because the coefficient of $(x + 2)^2$ is positive.
• The axis of symmetry is $x = -2$.

Both graphs represent parabolas that open upwards, but their vertices are different due to horizontal and vertical shifts.

Would you like a graphical representation or more details on how to draw the graph on the grid?

5 Questions for Expansion:

1. How do we determine the direction of a parabola (up or down)?
2. What is the effect of changing the sign of the square term, for example $y = -(x - 3)^2$?
3. How can we find the points where these parabolas intersect the x-axis or y-axis?
4. What are some applications of quadratic functions like these in real-world problems?
5. How would the graph change if we added a coefficient in front of the squared term, like $y = 2(x - 3)^2$?

Tip:

The vertex form of a quadratic function is useful for easily identifying the vertex and direction of the parabola, making graphing simpler.