Opening Down
Part of the Quadratics & Complex Numbers section of Coddy's Algebra 2 journey. Lesson 15 of 51.
A parabola that opens down has a negative a, often just a minus in front: -x^2 + 2x + 3. Treat that minus as a = -1 and take it out of the two x terms: -(x^2 - 2x) + 3. Every sign inside flips, so the +2x becomes -2x.
Complete the square inside: x^2 - 2x is (x - 1)^2 - 1, so the expression is -((x - 1)^2 - 1) + 3. Expanding the minus flips the -1 to +1: -(x - 1)^2 + 1 + 3, which is -(x - 1)^2 + 4.
The vertex is (1, 4), and since a is -1 it is the highest point: 4 is the largest value the expression ever takes. Check it with the axis formula: -2 over 2 times -1 is 1, and putting 1 into the original gives -1 + 2 + 3, which is 4.
Challenge
Write -x^2 + 6x - 4 in vertex form.
Try it yourself
Write it in vertex form
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Quadratics & Complex Numbers
2Completing the Square
Perfect Square TrinomialsThe Missing NumberCompleting the SquareIrrational AnswersWhen a Is Not 1Recap: Completing the Square5Complex Arithmetic
Adding and SubtractingMultiplyingThe ConjugateDividingA Pure Imaginary BottomRecap: Complex Arithmetic8Lines Meet Parabolas
Two CrossingsJust TouchingNo MeetingA Circle and a LineRecap: Lines and Parabolas3Vertex Form by Algebra
From Standard to VertexOdd Middle TermsA Leading NumberOpening DownRecap: Vertex Form