Question: Question 6 The function $f$ is defined for all real values of $x$ as $f(x) = c + 8x - x^2$, where Given that the range of $f$ is $f(x) \leq 19$, find the value of $c$. Graph: a downward-opening parabola for $f(x) = c + 8x - x^2$ with vertex at $(4, c + 16)$; position_hint = center User: Solve

1. **State the problem:** We have a quadratic function defined as $$f(x) = c + 8x - x^2$$ and we know the range is $$f(x) \leq 19$$. We need to find the value of the constant $$c$$. 2. **Identify the vertex form and vertex:** The quadratic can be rewritten to find the vertex. The coefficient of $$x^2$$ is $$-1$$, so the parabola opens downward. The vertex $$x$$-coordinate is given by $$x = -\frac{b}{2a}$$ where $$a = -1$$ and $$b = 8$$. Calculate: $$x = -\frac{8}{2 \times (-1)} = -\frac{8}{-2} = 4$$ 3. **Find the vertex $$y$$-coordinate:** Substitute $$x=4$$ into $$f(x)$$: $$f(4) = c + 8(4) - (4)^2 = c + 32 - 16 = c + 16$$ 4. **Use the range condition:** Since the parabola opens downward, the vertex represents the maximum value of $$f(x)$$. Given the range $$f(x) \leq 19$$, the maximum value is 19, so: $$c + 16 = 19$$ 5. **Solve for $$c$$:** $$c = 19 - 16 = 3$$ **Final answer:** $$\boxed{3}$$