Let's break down the problem: **Original equation:** \( x^2 + y^2 = 9 \) This is a circle centered at (0, 0) with radius 3. **Transformations:** - Up 3 units: *y* becomes *(y - 3)* - Left 4 units: *x* becomes *(x + 4)* **Equation of shifted graph:** Replace \( x \) with \( x + 4 \) and \( y \) with \( y - 3 \): \[ (x + 4)^2 + (y - 3)^2 = 9 \] So, the blank should be filled with: \[ (x + 4)^2 + (y - 3)^2 \] **Now, which graph matches?** - The original circle is at (0, 0). - The shifted circle should be at (-4, 3). Let's check the options: - **A:** Shift is right and down (incorrect). - **B:** Shift is right and up (incorrect). - **C:** Shift is left and up (correct!). - **D:** Shift is left and down (incorrect). **Final answers:** - The equation of the shifted graph is: \((x + 4)^2 + (y - 3)^2 = 9\) - The correct graph is: **C**

Let's analyze and solve this problem step by step. --- ### Step 1: Understand the original graph The original equation is: \[ x^2 + y^2 = 9 \] This is a **circle** centered at **(0, 0)** with **radius 3**. --- ### Step 2: Identify the transformations The problem states: - The graph is shifted **up 3 units**. - The graph is shifted **left 4 units**. --- ### Step 3: Write the transformation equations **Vertical shift (up 3 units):** replace \( y \) with \( y - 3 \) **Horizontal shift (left 4 units):** replace \( x \) with \( x + 4 \) --- ### Step 4: Write the equation of the shifted graph Apply the shifts to the original equation: \[ (x + 4)^2 + (y - 3)^2 = 9 \] This is the equation of the **shifted circle**. --- ### Step 5: Determine the new center - Original center: **(0, 0)** - After shifting **left 4 units**: center moves to **(-4, 0)** - After shifting **up 3 units**: center moves to **(-4, 3)** So, the new circle has: - Center at **(-4, 3)** - Radius remains **3** --- ### Step 6: Identify the correct graph - The original circle is at (0,0) - The shifted circle is at (-4, 3) From the options: - **A:** Shift is right and down (incorrect) - **B:** Shift is right and up (incorrect) - **C:** Shift is left and up (**correct**) - **D:** Shift is left and down (incorrect) --- ### **Final answer:** - **Equation of the shifted circle:** \[ \boxed{(x + 4)^2 + (y - 3)^2 = 9} \] - **Graph representation:** The correct graph is **option C**, which shows the circle shifted **left 4 units** and **up 3 units** from the original. --- **Note:** To sketch: - Draw the original circle at (0, 0). - Draw the shifted circle at (-4, 3). --- ### Summary: | Step | Description | Result | |---|---|---| | 1 | Original equation | \(x^2 + y^2 = 9\) | | 2 | Shifts: up 3, left 4 | \( (x + 4)^2 + (y - 3)^2 = 9 \) | | 3 | New center | \((-4, 3)\) | | 4 | Correct graph choice | **Option C** | --- Let me know if you'd like to see a sketch or further explanation!