Graphs and transformations: MI 2024 Promo Q3

(i) [4] $$y=ax^3+bx^2+cx+d$$ At $(0,7)$: $d=7$.

At $(-2,-11)$: $$\begin{aligned} -11&=a(-2)^3+b(-2)^2+c(-2)+7\\ -8a+4b-2c&=-18\quad\cdots\cdots(1) \end{aligned}$$ At $(1,1)$: $$\begin{aligned} 1&=a(1)^3+b(1)^2+c(1)+7\\ a+b+c&=-6\quad\cdots\cdots(2) \end{aligned}$$ At $(3,19)$: $$\begin{aligned} 19&=a(3)^3+b(3)^2+c(3)+7\\ 27a+9b+3c&=12\quad\cdots\cdots(3) \end{aligned}$$ Using GC, $a=2$, $b=-3$ and $c=-5$.

3(ii) [1]

Equation of $C$: $y=2x^3-3x^2-5x+7$. $$y=2x^3-3x^2-5x+7 \quad\xrightarrow{\text{replace $x$ with $-x$}}\quad y=2(-x)^3-3(-x)^2-5(-x)+7$$ $$y-1=2(-x)^3-3(-x)^2-5(-x)+7$$ $$\therefore\quad y=-2x^3-3x^2+5x+8$$