Name:_________________
January 28, 2014
Algebra 2
Quadratic Functions: Putting it all Together
We’ve learned about three forms for quadratic functions, and we’ve covered all of the skills
needed to convert between the forms (distributing, factoring, and completing the square). You
can now find enough information about a quadratic function (vertex, zeros, y-intercept) to be
able to sketch the graph without a calculator.
Three forms for quadratic functions
Here are three general forms that can be used for writing formulas for quadratic functions.
Each has its advantages.
form
standard
factored
vertex
function formula
f(x) = ax2 + bx + c
f(x) = (px + q)(rx + s)
main advantage
ready for using the Quadratic Formula
Find zeros by solving the equations
px + q = 0 and rx + s = 0.
OR
OR
f (x) = a(x - x1)(x - x 2 ) x1 and x 2 are the zeros
f(x) = a(x – h)2 + k
The vertex is (h, k).
Equation Solving Methods (Quiz on this tomorrow)
 Factoring (to use: put into factored form)
 Quadratic Formula (to use: put into standard form)
 Completing the Square (to use: start with standard form)
 Square Roots (to use: put into vertex form)
Converting between the forms
You’ve already learned all of the skills needed to change a quadratic function from any of the
forms to another form. Specifically here’s what’s needed in each case:
conversion
standard to factored
standard to vertex
factored to standard
vertex to standard
how to do it
factoring, and maybe some extra steps
completing the square OR find the vertex
distributing (mult. table) and simplifying (combine
like terms)
distributing (mult. table) and simplifying (combine
like terms)
To get back and forth between factored and vertex forms, use standard form as an
in-between step.
Example: Convert f(x) = 3(x – 6)(x – 2) into vertex form.
Steps to get from factored to standard form
First multiply (x – 6)(x – 2):
f(x) = 3(x2 – 8x + 12)
Distribute the 3:
f(x) = 3x2 – 24x + 36
Steps to get from standard to vertex form (easier way):
 b   24 24
Use the formula to find the x component of the vertex: x 


4
2a
23
6
Plug in to find y = 3(4)2 – 24(4) +36 = −12
Write the equation:
f(x) = 3 (x – 4)2 – 12
Name:_________________
January 28, 2014
Algebra 2
Problems: Converting
1. Change each function into the form specified. If you’re not sure what to do, see the chart on
page 1.
a. f(x) = x2 – 4x – 96 into factored form.
b. f(x) = 4x2 – 4x – 3 into factored form.
c. f(x) = 3(x – 4)(x + 2) into standard form.
Hint: First do (x – 4)(x + 2)
then use the 3.
d. f(x) = –2 (x + 5)2 + 6 into standard form.
Hint: First do (x + 5)2,
then use the –2, then the 6.
Name:_________________
January 28, 2014
Algebra 2
e. f(x) = 2x2 + 16x + 28 into vertex form.
f. f(x) = (x + 3)(x – 5) into vertex form.
Hint: First distribute, then
completing-the-square.
g. f(x) = (x – 1)2 – 1 into factored form.
Hint: Distribute, then
simplify, then factor.
h. f(x) = 3(x – 4)(x + 2) into vertex form.
Name:_________________
January 28, 2014
i. f(x) = 2(x – 3)2 – 8 into factored form.
j. f(x) = –(x – 5)2 – 3 into standard form.
k. f(x) = –2x2 – 12x – 18 into factored form.
l. f(x) = –2x2 – 12x – 18 into vertex form.
Algebra 2