Olympiad Maths Prep

Track / Stage 4 / 157 of 340 #417 of 2000

Problem 417

AMC 12 late, AIME early
Geometry Difficulty 4.8 Prove it 2. matematično tekmovanje dijakov srednjih tehniških in strokovnih šol · Slovenia

Problem:

Nariši graf funkcije s predpisom f(x)=−x2−6x+9f(x) = -\sqrt{x^{2} - 6x + 9} in izračunaj ploščino lika, ki ga oklepa graf dane funkcije s koordinatnima osema.

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

Solution:

Zapis f(x)=−(x−3)2f(x) = -\sqrt{(x-3)^{2}}

Zapis f(x)=−∣x−3∣f(x) = -|x-3|

Izračunana ničla (m)(m), začetna vrednost (n):m=3,n=−3(n): m = 3, n = -3

Narisan graf funkcije g(x)=−(x−3)g(x) = -(x-3)

Narisan graf funkcije f(x)=−∣x−3∣f(x) = -|x-3|

Izračunana ploščina trikotnika S=92S = \frac{9}{2}

Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.