1. **Problem:** Transform the parabola equation from $x^2 = 2y + 11$ to $2(x - 1)^2 = 2y + 5$ and verify if the point $(-5,7)$ lies on the new parabola. 2. **Step 1: Understand the original equation** The original parabola is given by: $$x^2 = 2y + 11$$ This can be rearranged to express $y$: $$2y = x^2 - 11 \implies y = \frac{x^2 - 11}{2}$$ 3. **Step 2: Understand the new equation** The new equation is: $$2(x - 1)^2 = 2y + 5$$ Rearranged to solve for $y$: $$2y = 2(x - 1)^2 - 5 \implies y = (x - 1)^2 - \frac{5}{2}$$ 4. **Step 3: Verify the point $(-5,7)$ on the new parabola** Substitute $x = -5$ and $y = 7$ into the new equation: $$2(x - 1)^2 = 2y + 5$$ Calculate left side: $$2(-5 - 1)^2 = 2(-6)^2 = 2 \times 36 = 72$$ Calculate right side: $$2 \times 7 + 5 = 14 + 5 = 19$$ Since $72 \neq 19$, the point $(-5,7)$ does not lie on the new parabola. 5. **Summary:** - Original parabola: $y = \frac{x^2 - 11}{2}$ - New parabola: $y = (x - 1)^2 - \frac{5}{2}$ - Point $(-5,7)$ is not on the new parabola because substituting it does not satisfy the equation. Final answer: The point $(-5,7)$ does not lie on the parabola defined by $2(x - 1)^2 = 2y + 5$.