1. **Problem statement:** We have a parabola $p$ with vertex $S(3|5)$ and equation $y = ax^2 + bx + c$ where $a,b,c \in \mathbb{R}$. Also, a line $g$ with equation $y = -0.25x + 3$. You want to understand each subproblem clearly and simply. 2. **B 1.1: Show that the parabola has equation $y = -0.5x^2 + 3x + 0.5$** - Use the vertex form of a parabola: $$y = a(x - h)^2 + k$$ where $(h,k)$ is the vertex. - Here, $h=3$, $k=5$, so $$y = a(x - 3)^2 + 5$$ - Expand: $$y = a(x^2 - 6x + 9) + 5 = a x^2 - 6a x + 9a + 5$$ - Compare with $y = ax^2 + bx + c$, so $b = -6a$, $c = 9a + 5$. - To find $a$, use another point on the parabola or conditions given. - The problem states the parabola equation is $y = -0.5x^2 + 3x + 0.5$, so check if $b = 3$ and $c = 0.5$ match the above: - $b = -6a = 3 \Rightarrow a = -0.5$ - $c = 9a + 5 = 9(-0.5) + 5 = -4.5 + 5 = 0.5$ - So the equation is confirmed. 3. **B 1.2: Points $B$ on line $g$ and $D$ on parabola $p$ share the same $x$-coordinate. Together with points $A$ and $C$, they form kite-shaped quadrilaterals with symmetry axes and diagonals.** - Understand that $B$ and $D$ depend on $x$. - $B = (x, -0.25x + 3)$, $D = (x, -0.5x^2 + 3x + 0.5)$. - $A$ and $C$ are fixed or defined points. - The kite shape means two pairs of adjacent sides are equal and there is a symmetry axis. - You will use these points to analyze the kite properties. 4. **B 1.3: Explain why the area of triangle $A,B,P$ is always half the area of triangle $B,C,D_n$.** - Use the formula for the area of a triangle given coordinates. - Show the ratio of areas by comparing base and height or using coordinate geometry. - This is a property of the kite and the points chosen. 5. **B 1.4: Find for which $x$ values kite-shaped quadrilaterals $A,B,C,D_n$ exist.** - Use the conditions for a kite: equal adjacent sides and symmetry. - Set equations for side lengths equal and solve for $x$. - This will give the domain of $x$ where the kite exists. 6. **B 1.5: Among these kites, the one with points $A,B,C,D$ has the maximum area. Calculate this area and the corresponding $x$.** - Express the area as a function of $x$. - Use calculus or vertex formula for quadratic functions to find maximum. - Calculate the maximum area and the $x$ value. **Tips for your exam:** - Always write down what is given and what you need to find. - Use formulas step-by-step and explain each step. - Check your units and decimal places (two decimals as requested). - Practice simplifying expressions and canceling terms carefully. - For speed, practice recognizing vertex form and area formulas. - Don’t forget to verify your answers by plugging back into equations. - Keep calm and write clearly. Good luck! You can do it!