Question: Activity 2: Sketch My Graph 1) y \geq x^2 - 6x + 5 2) y < 2x^2 - 3x + 1 Graph 1: upward-opening parabola for y = x^2 - 6x + 5, vertex at (3, -4), solid boundary, shade above; position_hint=top-left. Graph 2: upward-opening parabola for y = 2x^2 - 3x + 1, vertex at (3/4, -1/8), dashed boundary, shade below; position_hint=top-left. Activity 1: Do I Belong Here? Write YES if the point is a solution of the inequality. Otherwise, write NO. In addition, ask them to provide a short justification of their answers. The items are the following: 1) y > x^2 - 7x - 9 (4, -2) 2) y < x(x + 6) + 5 (-6, 1/2) 3) y \geq 5x^2 - 17x + 6 (2, -6) 4) y - 5 > 2x^2 + 11x (-4, 5/2) 5) y \leq x(3x-10) + 3 (1, 8)

1. **Problem:** Determine if the point $(4, -2)$ satisfies the inequality $y > x^2 - 7x - 9$. 2. **Formula:** Substitute $x=4$ and $y=-2$ into the inequality. 3. **Calculation:** $$x^2 - 7x - 9 = 4^2 - 7\times4 - 9 = 16 - 28 - 9 = -21$$ 4. **Check inequality:** $$-2 > -21$$ This is true, so the answer is **YES**. 1. **Problem:** Determine if the point $(-6, \frac{1}{2})$ satisfies $y < x(x + 6) + 5$. 2. **Formula:** Substitute $x=-6$, $y=\frac{1}{2}$. 3. **Calculation:** $$x(x+6)+5 = (-6)(-6+6)+5 = (-6)(0)+5 = 5$$ 4. **Check inequality:** $$\frac{1}{2} < 5$$ True, so **YES**. 1. **Problem:** Determine if $(2, -6)$ satisfies $y \geq 5x^2 - 17x + 6$. 2. **Formula:** Substitute $x=2$, $y=-6$. 3. **Calculation:** $$5(2)^2 - 17(2) + 6 = 5\times4 - 34 + 6 = 20 - 34 + 6 = -8$$ 4. **Check inequality:** $$-6 \geq -8$$ True, so **YES**. 1. **Problem:** Determine if $(-4, \frac{5}{2})$ satisfies $y - 5 > 2x^2 + 11x$. 2. **Formula:** Substitute $x=-4$, $y=\frac{5}{2}$. 3. **Calculation:** $$y - 5 = \frac{5}{2} - 5 = \frac{5}{2} - \frac{10}{2} = -\frac{5}{2}$$ $$2x^2 + 11x = 2(-4)^2 + 11(-4) = 2\times16 - 44 = 32 - 44 = -12$$ 4. **Check inequality:** $$-\frac{5}{2} > -12$$ Since $-2.5 > -12$ is true, answer is **YES**. 1. **Problem:** Determine if $(1, 8)$ satisfies $y \leq x(3x-10) + 3$. 2. **Formula:** Substitute $x=1$, $y=8$. 3. **Calculation:** $$x(3x-10) + 3 = 1(3\times1 - 10) + 3 = 1(-7) + 3 = -7 + 3 = -4$$ 4. **Check inequality:** $$8 \leq -4$$ False, so answer is **NO**. **Summary:** 1) YES 2) YES 3) YES 4) YES 5) NO Please provide a short justification for each answer as shown in the steps above.