1. **State the problem:** We need to determine which points lie on the graph of the transformed function $$y = f(x + 7) - 3$$ given points and the original function $$f(x)$$. 2. **Recall the transformation rules:** - The transformation $$f(x + 7)$$ shifts the graph of $$f(x)$$ 7 units to the left. - The transformation $$-3$$ outside the function shifts the graph down by 3 units. 3. **Use the point transformation rule:** If a point $$(a, b)$$ lies on $$y = f(x)$$, then the corresponding point on $$y = f(x + 7) - 3$$ is $$(a, f(a + 7) - 3)$$. 4. **Check each point:** - For $$(7, -3)$$: Check if $$-3 = f(7 + 7) - 3 = f(14) - 3$$, so $$f(14) = 0$$. - For $$(6, -2)$$: Check if $$-2 = f(6 + 7) - 3 = f(13) - 3$$, so $$f(13) = 1$$. - For $$(-7, -3)$$: Check if $$-3 = f(-7 + 7) - 3 = f(0) - 3$$, so $$f(0) = 0$$. - For $$(-6, -2)$$: Check if $$-2 = f(-6 + 7) - 3 = f(1) - 3$$, so $$f(1) = 1$$. - For $$(-3, -7)$$: Check if $$-7 = f(-3 + 7) - 3 = f(4) - 3$$, so $$f(4) = -4$$. - For $$(-2, -6)$$: Check if $$-6 = f(-2 + 7) - 3 = f(5) - 3$$, so $$f(5) = -3$$. 5. **Use the given graph information:** - The parabola is upward-opening with vertex at $$(0,0)$$. - The point $$(1,1)$$ is on $$f(x)$$. 6. **Evaluate the points:** - $$f(0) = 0$$ (vertex), so $$(-7, -3)$$ is on the transformed graph. - $$f(1) = 1$$, so $$(-6, -2)$$ is on the transformed graph. - $$f(14)$$ and $$f(13)$$ are not given, so cannot confirm $$(7, -3)$$ or $$(6, -2)$$. - $$f(4)$$ and $$f(5)$$ are not given and likely positive for an upward parabola, so $$( -3, -7)$$ and $$(-2, -6)$$ are unlikely. **Final answer:** The points on the graph of $$y = f(x + 7) - 3$$ are $$(-7, -3)$$ and $$(-6, -2)$$.