1. **Problem 1: Evaluate the difference quotient for** $f(x) = x^2 - 2x + 3$. 2. The difference quotient is given by the formula: $$\frac{f(a+h) - f(a)}{h}$$ where $a$ and $h$ are variables and $h \neq 0$. 3. First, find $f(a+h)$ by substituting $x = a+h$ into $f(x)$: $$f(a+h) = (a+h)^2 - 2(a+h) + 3$$ Expand and simplify: $$= a^2 + 2ah + h^2 - 2a - 2h + 3$$ 4. Next, find $f(a)$: $$f(a) = a^2 - 2a + 3$$ 5. Compute the numerator of the difference quotient: $$f(a+h) - f(a) = (a^2 + 2ah + h^2 - 2a - 2h + 3) - (a^2 - 2a + 3)$$ Simplify by canceling terms: $$= 2ah + h^2 - 2h$$ 6. Divide by $h$: $$\frac{2ah + h^2 - 2h}{h} = \frac{h(2a + h - 2)}{h}$$ Cancel $h$ (since $h \neq 0$): $$= 2a + h - 2$$ 7. **Final answer:** $$\frac{f(a+h) - f(a)}{h} = 2a + h - 2$$ --- 8. **Problem 2: Which rules define functions based on the given graphs?** 9. A function is defined such that for each input $x$, there is exactly one output $y$. 10. The **Vertical Line Test** states: If any vertical line intersects the graph more than once, the graph does not represent a function. 11. Analyze the graphs: - Top row: 1. Parabola opening upwards: passes vertical line test, so it is a function. 2. Straight line through origin: passes vertical line test, so it is a function. 3. Curve with one local max and min: passes vertical line test, so it is a function. 4. Semicircle: passes vertical line test, so it is a function. - Bottom row: 1. Circle centered at origin: vertical lines intersect twice, so not a function. 2. Vertical wave-like curve: vertical lines intersect once or multiple times depending on shape, but since it is vertical wave-like, vertical lines may intersect multiple times, so not a function. 3. Vertical straight line at $x=0$: vertical line test fails (infinite outputs for $x=0$), so not a function. 4. Sideways figure eight: vertical lines intersect twice, so not a function. 12. **Summary:** The vertical line test is the rule that defines whether a graph represents a function.