Writing Quadratic Equations Notes
Writing Quadratic Equations from Points
Vertex Form \(y = a(x - h)^2 + k\)
Given vertex \((h, k)\) and a random point \((x, y)\)
Intercept Form \(y = a(x - r_1)(x - r_2)\)
Given roots \((r_1, 0)\), \((r_2, 0)\) and a random point \((x, y)\)
Example 1
Vertex \((1, -2)\) and Random Point \((3, 6)\)
x y (1, -2) (3, 6)
<table class="font-mono text-[12px] border-collapse w-full"><tbody><tr class="text-slate-500 text-[11px]"><td class="text-right pr-2 py-0.5 whitespace-nowrap">y</td><td class="text-center px-1.5 font-semibold py-0.5 whitespace-nowrap">=</td><td class="text-left pl-2 py-0.5 whitespace-nowrap">a(x − h)² + k</td><td class="text-left pl-3 text-slate-400 text-[10.5px] font-sans italic whitespace-nowrap">Vertex form</td></tr><tr class="font-semibold text-slate-900"><td class="text-right pr-2 py-0.5 whitespace-nowrap">(6)</td><td class="text-center px-1.5 py-0.5 whitespace-nowrap">=</td><td class="text-left pl-2 py-0.5 whitespace-nowrap">a((3) − (1))² + (−2)</td><td class="text-left pl-3 text-slate-400 text-[10.5px] font-sans italic whitespace-nowrap">Substitute (1, -2) and (3, 6)</td></tr><tr><td class="text-right pr-2 py-0.5 whitespace-nowrap">6</td><td class="text-center px-1.5 py-0.5 whitespace-nowrap">=</td><td class="text-left pl-2 py-0.5 whitespace-nowrap">a(2)² − 2</td><td class="text-left pl-3 text-slate-400 text-[10.5px] font-sans italic whitespace-nowrap">Simplify (3 − 1 = 2)</td></tr><tr><td class="text-right pr-2 py-0.5 whitespace-nowrap">6</td><td class="text-center px-1.5 py-0.5 whitespace-nowrap">=</td><td class="text-left pl-2 py-0.5 whitespace-nowrap">4a − 2</td><td class="text-left pl-3 text-slate-400 text-[10.5px] font-sans italic whitespace-nowrap">Evaluate 2² = 4</td></tr><tr class="text-slate-400 font-bold text-[11px]"><td class="text-right pr-2 pt-0 pb-0.5 whitespace-nowrap">+2</td><td class="text-center px-1.5 pt-0 pb-0.5 whitespace-nowrap"></td><td class="text-left pl-2 pt-0 pb-0.5 whitespace-nowrap"><span class="opacity-0 select-none">4a </span>+2</td><td class="text-left pl-3 text-slate-400 text-[10px] font-sans font-normal italic whitespace-nowrap">Add 2 to both sides</td></tr><tr class="font-semibold text-slate-900 border-t border-slate-300"><td class="text-right pr-2 pt-0.5 pb-0 whitespace-nowrap">8</td><td class="text-center px-1.5 pt-0.5 pb-0 whitespace-nowrap">=</td><td class="text-left pl-2 pt-0.5 pb-0 whitespace-nowrap">4a</td><td class="text-left pl-3 pt-0.5 pb-0 text-slate-400 text-[10.5px] font-sans italic whitespace-nowrap">Combine like terms</td></tr><tr class="text-slate-400 font-bold text-[11px]"><td class="text-right pr-2 pt-0 pb-0.5 whitespace-nowrap"><div class="inline-flex flex-col items-center"><div class="w-5 border-t border-slate-400"></div><span class="text-slate-400 text-[10.5px] leading-tight">4</span></div></td><td class="text-center px-1.5 pt-0 pb-0.5 whitespace-nowrap"></td><td class="text-left pl-2 pt-0 pb-0.5 whitespace-nowrap"><div class="inline-flex flex-col items-center"><div class="w-6 border-t border-slate-400"></div><span class="text-slate-400 text-[10.5px] leading-tight">4</span></div></td><td class="text-left pl-3 text-slate-400 text-[10px] font-sans font-normal italic whitespace-nowrap">Divide both sides by 4</td></tr><tr class="font-bold text-blue-900 border-t border-slate-300"><td class="text-right pr-2 py-0.5 whitespace-nowrap">2</td><td class="text-center px-1.5 py-0.5 whitespace-nowrap">=</td><td class="text-left pl-2 py-0.5 whitespace-nowrap">a → a = 2</td><td class="text-left pl-3 py-0.5 text-slate-400 text-[10.5px] font-sans italic whitespace-nowrap">Solved value of a</td></tr></tbody></table>
Equation: y = 2(x − 1)² − 2
Example 2
Roots \((-1, 0)\), \((5, 0)\) and Random Point \((1, 8)\)
x y (-1, 0) (5, 0) (1, 8)
<table class="font-mono text-[12px] border-collapse w-full"><tbody><tr class="text-slate-500 text-[11px]"><td class="text-right pr-2 py-0.5 whitespace-nowrap">y</td><td class="text-center px-1.5 font-semibold py-0.5 whitespace-nowrap">=</td><td class="text-left pl-2 py-0.5 whitespace-nowrap">a(x − r₁)(x − r₂)</td><td class="text-left pl-3 text-slate-400 text-[10.5px] font-sans italic whitespace-nowrap">Intercept form</td></tr><tr class="font-semibold text-slate-900"><td class="text-right pr-2 py-0.5 whitespace-nowrap">(8)</td><td class="text-center px-1.5 py-0.5 whitespace-nowrap">=</td><td class="text-left pl-2 py-0.5 whitespace-nowrap">a((1) − (−1))((1) − (5))</td><td class="text-left pl-3 text-slate-400 text-[10.5px] font-sans italic whitespace-nowrap">Substitute roots & (1, 8)</td></tr><tr><td class="text-right pr-2 py-0.5 whitespace-nowrap">8</td><td class="text-center px-1.5 py-0.5 whitespace-nowrap">=</td><td class="text-left pl-2 py-0.5 whitespace-nowrap">a(1 + 1)(−4)</td><td class="text-left pl-3 text-slate-400 text-[10.5px] font-sans italic whitespace-nowrap">Simplify 1 − (−1) = 1 + 1</td></tr><tr><td class="text-right pr-2 py-0.5 whitespace-nowrap">8</td><td class="text-center px-1.5 py-0.5 whitespace-nowrap">=</td><td class="text-left pl-2 py-0.5 whitespace-nowrap">a(2)(−4)</td><td class="text-left pl-3 text-slate-400 text-[10.5px] font-sans italic whitespace-nowrap">Simplify (1 + 1 = 2)</td></tr><tr><td class="text-right pr-2 py-0.5 whitespace-nowrap">8</td><td class="text-center px-1.5 py-0.5 whitespace-nowrap">=</td><td class="text-left pl-2 py-0.5 whitespace-nowrap">−8a</td><td class="text-left pl-3 text-slate-400 text-[10.5px] font-sans italic whitespace-nowrap">Multiply 2 × (−4) = −8</td></tr><tr class="text-slate-400 font-bold text-[11px]"><td class="text-right pr-2 pt-0 pb-0.5 whitespace-nowrap"><div class="inline-flex flex-col items-center"><div class="w-5 border-t border-slate-400"></div><span class="text-slate-400 text-[10.5px] leading-tight">−8</span></div></td><td class="text-center px-1.5 pt-0 pb-0.5 whitespace-nowrap"></td><td class="text-left pl-2 pt-0 pb-0.5 whitespace-nowrap"><div class="inline-flex flex-col items-center"><div class="w-7 border-t border-slate-400"></div><span class="text-slate-400 text-[10.5px] leading-tight">−8</span></div></td><td class="text-left pl-3 text-slate-400 text-[10px] font-sans font-normal italic whitespace-nowrap">Divide both sides by −8</td></tr><tr class="font-bold text-emerald-900 border-t border-slate-300"><td class="text-right pr-2 py-0.5 whitespace-nowrap">−1</td><td class="text-center px-1.5 py-0.5 whitespace-nowrap">=</td><td class="text-left pl-2 py-0.5 whitespace-nowrap">a → a = −1</td><td class="text-left pl-3 py-0.5 text-slate-400 text-[10.5px] font-sans italic whitespace-nowrap">Solved value of a</td></tr></tbody></table>
Equation: y = −(x + 1)(x − 5)
