1. **State the problem:** We want to understand how changing the value of $c$ in the quadratic function $y = ax^2 + bx + c$ affects the output values $y$ for different $x$ values. 2. **Recall the standard form:** The quadratic function is given by $$y = ax^2 + bx + c$$ where $a$, $b$, and $c$ are constants. Here, $a=1$, $b=0$, and $c$ varies. 3. **Effect of $c$:** Changing $c$ shifts the graph of the parabola vertically up or down without changing its shape. 4. **Calculate missing values:** Given $y = x^2 + 10$ and $y = x^2 - 3$, fill in the missing $y$ values for $x = -1, 0, 1, 2, 3$. For $y = x^2 + 10$: - At $x = -1$: $y = (-1)^2 + 10 = 1 + 10 = 11$ - At $x = 0$: $y = 0^2 + 10 = 0 + 10 = 10$ - At $x = 1$: $y = 1^2 + 10 = 1 + 10 = 11$ - At $x = 2$: $y = 2^2 + 10 = 4 + 10 = 14$ - At $x = 3$: $y = 3^2 + 10 = 9 + 10 = 19$ For $y = x^2 - 3$: - At $x = -1$: $y = (-1)^2 - 3 = 1 - 3 = -2$ - At $x = 0$: $y = 0^2 - 3 = 0 - 3 = -3$ - At $x = 1$: $y = 1^2 - 3 = 1 - 3 = -2$ - At $x = 2$: $y = 2^2 - 3 = 4 - 3 = 1$ - At $x = 3$: $y = 3^2 - 3 = 9 - 3 = 6$ 5. **Completed table:** | $x$ | $y = x^2$ | $y = x^2 + 10$ | $y = x^2 - 3$ | |-----|-----------|----------------|---------------| | -3 | 9 | 19 | 6 | | -2 | 4 | 14 | 1 | | -1 | 1 | 11 | -2 | | 0 | 0 | 10 | -3 | | 1 | 1 | 11 | -2 | | 2 | 4 | 14 | 1 | | 3 | 9 | 19 | 6 | 6. **Summary:** Changing $c$ shifts the parabola vertically. Increasing $c$ moves the graph up, decreasing $c$ moves it down, but the shape remains the same. **Final answer:** The missing values are filled as above, showing the effect of $c$ on the quadratic function.