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)
Writing Quadratic Equations Worksheet
Writing Quadratic Equations from Points
Name:
Directions: For each problem, determine the value of ‘\(a\)’ by substituting the given points into the specified form. Show all algebraic work vertically and write your final equation in the box provided.
Problem 1
Write the equation in vertex form: \(y = a(x - h)^2 + k\)
(1, 3) (3, -5)
Answer: \(y =\) ______________________
Problem 2
Write the equation in intercept form: \(y = a(x - r_1)(x - r_2)\)
(-2, 0) (4, 0) (0, -4)
Answer: \(y =\) ______________________
Problem 3
A parabola has its vertex located at \((3, -4)\) and passes through the random point \((1, 4)\).
Write the equation of the parabola in vertex form.
Answer: \(y =\) ______________________
Problem 4
A quadratic function has \(x\)-intercepts at \((-3, 0)\) and \((1, 0)\) and passes through the random point \((2, -5)\).
Write the equation of the quadratic function in intercept form.
Answer: \(y =\) ______________________
Writing Quadratic Equations from Points • Page 2
Problems 5 – 8
Problem 5
Write the equation of the parabola in vertex form.
(-2, -3) (0, 5)
Answer: \(y =\) ______________________
Problem 6
Write the equation of the quadratic function in intercept form.
(-1, 0) (3, 0) (2, 6)
Answer: \(y =\) ______________________
Problem 7
A parabola has its vertex at \((2, 5)\) and passes through the random point \((5, -4)\).
Write the equation of the parabola in vertex form.
Answer: \(y =\) ______________________
Problem 8 • Challenge
Determine the equation of the parabola below. Read coordinates from the grid to identify the vertex or intercepts, select a point, and solve for ‘\(a\)’.
0 1 2 3 4 1 -1 -4
Answer: \(y =\) ______________________
