Coordinate Geometry
Quadratic graphs with vertex relationship — x-intercept spread
MJAT_TS4_P1
Grade 12
Question:
Functions $f$ and $g$ are quadratic, $g(x)=-f(100-x)$, and the graph of $g$ contains the vertex of the graph of $f$. The four $x$-intercepts on the two graphs have $x$-coordinates $x_1<x_2<x_3<x_4$, with $x_3-x_2=150$. The value of $x_4-x_1$ is $m+n\sqrt{p}$ where $m$, $n$, $p$ are positive integers and $p$ is square-free. Then $m-n+p=$
Step-by-Step Solution
Key Concept: Let $f(x)=a(x-h)^2+k$ (vertex $(h,k)$). Since $g(x)=-f(100-x)=-a(100-x-h)^2-k$: vertex of $g$ is at $(100-h,-k)$. Condition: graph of $g$ contains vertex of $f$, i.e., $g(h)=k$. $g(h)=-f(100-h)=k\Rightarrow f(100-h)=-k\Rightarrow a(100-2h)^2+k=-k\Rightarrow a(100-2h)^2=-2k$.
$m-n+p=450-300+2=\mathbf{152}$.
Correct Answer: 152