Let's solve this by matching each given equation to the appropriate graph based on their transformations. The general form of the quadratic equations in vertex form is:

$y = a(x - h)^2 + k$

Where $(h, k)$ is the vertex of the parabola, and $a$ determines the direction and width of the parabola.

Step 1: Analyze the given equations

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

   • Vertex: $(h, k) = (1, -4)$
   • This means the graph is shifted right by 1 and down by 4.

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

   • Vertex: $(h, k) = (2, 2)$
   • This means the graph is shifted right by 2 and up by 2.

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

   • Vertex: $(h, k) = (-2, 2)$
   • This means the graph is shifted left by 2 and up by 2.

4. Equation (d): $y = (x + 3)^2 - 2$

   • Vertex: $(h, k) = (-3, -2)$
   • This means the graph is shifted left by 3 and down by 2.

Step 2: Analyze the graphs

• Position 1: The vertex is at $(2, 2)$.
• Position 2: The vertex is at $(-2, 2)$.
• Position 3: The vertex is at $(-3, -2)$.
• Position 4: The vertex is at $(1, -4)$.

Step 3: Match the equations to the graphs

• Position 1 (vertex $(2, 2)$): This matches equation (b) $y = (x - 2)^2 + 2$.
• Position 2 (vertex $(-2, 2)$): This matches equation (c) $y = (x + 2)^2 + 2$.
• Position 3 (vertex $(-3, -2)$): This matches equation (d) $y = (x + 3)^2 - 2$.
• Position 4 (vertex $(1, -4)$): This matches equation (a) $y = (x - 1)^2 - 4$.

Final Answer:

• Position 1: $y = (x - 2)^2 + 2$ $\rightarrow$ (b)
• Position 2: $y = (x + 2)^2 + 2$ $\rightarrow$ (c)
• Position 3: $y = (x + 3)^2 - 2$ $\rightarrow$ (d)
• Position 4: $y = (x - 1)^2 - 4$ $\rightarrow$ (a)

Would you like further details on these transformations?

Relative Questions:

1. What does the coefficient $a$ in the equation do to the shape of the parabola?
2. How would the graph change if the $h$ value was negative in an equation?
3. Can you explain how the vertex form helps in identifying shifts of parabolas?
4. What is the difference between the standard form and the vertex form of a quadratic function?
5. How can you determine the axis of symmetry from the vertex form?

Tip:

Always use the vertex form of a quadratic function to easily determine the vertex and shifts along the x- and y-axis.