1. **State the problem:** We are given the function $f(x) = -x^2 - 5x + 2$ and need to find various difference quotients. 2. **Difference Quotient Definition:** The difference quotient generally refers to expressions like $$\frac{f(x+h) - f(x)}{h}$$ and similar versions where increments or points vary. 3. **Calculate $f(x+h)$:** $$f(x+h) = -(x+h)^2 - 5(x+h) + 2$$ $$= -(x^2 + 2xh + h^2) - 5x - 5h + 2$$ $$= -x^2 - 2xh - h^2 - 5x - 5h + 2$$ 4. **Compute the difference quotient $$\frac{f(x+h) - f(x)}{h}$$:** $$= \frac{(-x^2 - 2xh - h^2 - 5x - 5h + 2) - (-x^2 - 5x + 2)}{h}$$ $$= \frac{-x^2 - 2xh - h^2 - 5x - 5h + 2 + x^2 + 5x - 2}{h}$$ $$= \frac{-2xh - h^2 - 5h}{h}$$ $$= \frac{h(-2x - h - 5)}{h}$$ $$= -2x - h - 5$$ 5. **Interpretation:** The difference quotient simplifies to $$-2x - h - 5$$. 6. **Other common difference quotients:** - $$\frac{f(x) - f(a)}{x - a}$$ for a fixed number $a$. Calculate $f(a)$: $$f(a) = -a^2 - 5a + 2$$ Compute difference quotient: $$\frac{f(x) - f(a)}{x - a} = \frac{(-x^2 - 5x + 2) - (-a^2 - 5a + 2)}{x - a}$$ $$= \frac{-x^2 - 5x + 2 + a^2 + 5a - 2}{x - a}$$ $$= \frac{a^2 - x^2 + 5a - 5x}{x - a}$$ $$= \frac{(a - x)(a + x) + 5(a - x)}{x - a}$$ Note $a - x = -(x - a)$, so: $$= \frac{-(x - a)(a + x) - 5(x - a)}{x - a}$$ $$= - (a + x) - 5 = -a - x - 5$$ 7. **Summary:** - The difference quotient $$\frac{f(x+h) - f(x)}{h}$$ equals $$-2x - h - 5$$. - The difference quotient $$\frac{f(x) - f(a)}{x - a}$$ equals $$-a - x - 5$$. These represent the average rate of change between points on the parabola defined by $f(x)$.