1. **State the problem:** Given points $(-4,16)$, $(4,16)$, $(-5,25)$, and $(5,25)$, find the function that fits these points. 2. **Analyze the points:** Notice the points are symmetric about the y-axis, suggesting an even function, likely a quadratic or higher even power. 3. **Assume a quadratic function:** Let $y = ax^2 + bx + c$. Since the points are symmetric, $b=0$ (no linear term). 4. **Use points to find $a$ and $c$:** - Using $(4,16)$: $16 = a(4)^2 + c = 16a + c$ - Using $(5,25)$: $25 = a(5)^2 + c = 25a + c$ 5. **Solve the system:** $$\begin{cases} 16 = 16a + c \\ 25 = 25a + c \end{cases}$$ Subtract first from second: $$25 - 16 = 25a - 16a + c - c \Rightarrow 9 = 9a \Rightarrow a = 1$$ 6. **Find $c$:** $$16 = 16(1) + c \Rightarrow c = 0$$ 7. **Final function:** $$y = x^2$$ 8. **Verify:** - $(-4)^2 = 16$, $(4)^2 = 16$ - $(-5)^2 = 25$, $(5)^2 = 25$ All points satisfy $y = x^2$. **Answer:** The function is $y = x^2$.