Let's walk through the process of graphing these quadratic functions.

1. First Function: $y = (x + 5)^2 - 2$

This is in the vertex form $y = (x - h)^2 + k$, where $(h, k)$ is the vertex.

• Reflection: No reflection.

• Vertical Shift: Down 2 (since $-2$).

• Horizontal Shift: Left 5 (since $x + 5$).

• Vertical Stretch/Compression: None (the coefficient of the quadratic term is 1).

• Standard Form: Expand the quadratic: $(x + 5)^2 - 2 = x^2 + 10x + 25 - 2 = x^2 + 10x + 23.$

• Graphing: The parabola opens upwards (since the coefficient of $x^2$ is positive) and is shifted 5 units to the left and 2 units down. The vertex is at $(-5, -2)$.

2. Second Function: $y = (x - 2)^2 - 4$

This is also in vertex form:

• Reflection: No reflection.

• Vertical Shift: Down 4 (since $-4$).

• Horizontal Shift: Right 2 (since $x - 2$).

• Vertical Stretch/Compression: None.

• Standard Form: Expand the quadratic: $(x - 2)^2 - 4 = x^2 - 4x + 4 - 4 = x^2 - 4x.$

• Graphing: The parabola opens upwards, shifted 2 units to the right and 4 units down. The vertex is at $(2, -4)$.

3. Third Function: $y = -3(x - 2)^2 + 1$

This is in vertex form, but with a vertical reflection and stretch:

• Reflection: Yes, over the x-axis (since the coefficient is negative).

• Vertical Shift: Up 1 (since $+1$).

• Horizontal Shift: Right 2 (since $x - 2$).

• Vertical Stretch/Compression: Vertically stretched by a factor of 3 (since the coefficient of the quadratic term is -3).

• Standard Form: Expand the quadratic: $-3(x - 2)^2 + 1 = -3(x^2 - 4x + 4) + 1 = -3x^2 + 12x - 12 + 1 = -3x^2 + 12x - 11.$

• Graphing: The parabola opens downwards (due to the reflection) and is vertically stretched by 3. The vertex is at $(2, 1)$.

4. Fourth Function: $y = \frac{1}{3}(x - 4)^2 + 3$

This is also in vertex form:

• Reflection: No reflection.

• Vertical Shift: Up 3 (since $+3$).

• Horizontal Shift: Right 4 (since $x - 4$).

• Vertical Stretch/Compression: Vertically compressed by $\frac{1}{3}$ (since the coefficient is $\frac{1}{3}$).

• Standard Form: Expand the quadratic: $\frac{1}{3}(x - 4)^2 + 3 = \frac{1}{3}(x^2 - 8x + 16) + 3 = \frac{1}{3}x^2 - \frac{8}{3}x + \frac{16}{3} + 3 = \frac{1}{3}x^2 - \frac{8}{3}x + \frac{25}{3}.$

• Graphing: The parabola opens upwards, but is compressed vertically. The vertex is at $(4, 3)$.

Would you like more detailed steps or further clarification? Here are 5 follow-up questions for deeper understanding:

1. How does the vertex form of a quadratic equation help in graphing transformations?
2. How does vertical compression/stretch affect the shape of a parabola?
3. Why does a negative coefficient in front of $(x-h)^2$ cause a reflection over the x-axis?
4. How would you calculate the x-intercepts of these quadratic functions?
5. What are the differences in behavior between vertically compressed and stretched parabolas?

Tip:

When graphing quadratic equations, always start by identifying the vertex and shifts before determining the stretch/compression. This gives you a clearer view of the transformation effects!