Today's soundtrack is Christmas With the Mills Family, a Spotify playlist that includes my favourite Christmas songs.
Yesterday, I learned how to convert a quadratic function from general form to standard form if a > 0. This afternoon, I'll be working on the next step: converting when a < 0.
We use the following method to convert a quadratic function from general form to standard form if a < 0:
Remove the coefficient of a from both ax and bx as a common factor, making bx into a fraction if necessary, then put ax and bx inside of brackets, preceding the brackets with the coefficient of the removed value. Example: y = -4x² + 9x - 2 becomes y = -4(x² - 9/4x) - 2
Make a copy of the coefficient of the second term inside the brackets. Divide it by two, then square it. Inside our brackets, add and then subtract this new number. Example: -9/4 / 2 = -9/8; (-9/8)² = 81/64. This gives us y = -4(x² - 9/4x + 81/64 - 81/64) - 2
Rewrite the section inside the brackets as a perfect square, and inside a new set of brackets following the first set of brackets, multiply the fourth term by the same factor that precedes that first set of brackets. Example: y = -4(x² - 9/4x + 81/64 - 81/64) - 2 becomes y = -4(x² - 9/4x + 81/64) -4(-81/64) - 2
Convert the first set of brackets into a squared expression. Example: y = -4(x² - 9/4x + 81/64) -4(81/64) - 2 becomes y = -4(x - 9/8)² -4(-81/64) - 2
Multiply the second set of brackets by the constant preceding it Example: -81/64 ⋅ -4/1 = 324/64 = 81/16; therefore: y = -4(x - 9/8)² -4(81/64) - 2 becomes y = -4(x - 9/8)² + 81/16 - 2. 81/16 - 2/1 = 81/16 - 32/16 = 49/16; therefore: y = -4(x - 9/8)² + 81/16 - 2 becomes y = -4(x - 9/8)² + 49/16, our final answer! (Examples from Pearson's Pre-Calculus 11, p. 293)
Really, these steps are essentially identical to those in the lesson I worked on yesterday; I'm not really sure why they were presented as two different methods. Oh well. On to the homework portion I go!