Name:
Date: ______________
Algebra 2
Quadratic Functions: Putting it all Together page 1
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 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  x2 ) x1 and x2 are the zeros
f(x) = a(x – h)2 + k
The vertex is (h, k).
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
distributing (mult. table or “FOIL”) and
simplifying (combine like terms)
distributing (mult. table or “FOIL”) 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
Factor out 3 from first two terms:
Calculate: –8/2 = –4, (–4)2 = 16
Complete the square:
Write perfect square and simplify:
f(x) = 3(x2 – 8x
) + 36
f(x) = 3(x2 – 8x + 16) + 36 – 48
f(x) = 3 (x – 4)2
– 12
Name:
Date: ______________
Algebra 2
Quadratic Functions: Putting it all Together page 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:
Date: ______________
Algebra 2
Quadratic Functions: Putting it all Together page 3
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:
Date: ______________
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
Quadratic Functions: Putting it all Together page 4
Name:
Date: ______________
Algebra 2
Quadratic Functions: Putting it all Together page 5
Making Graphs by Hand
If you’re asked to graph of a quadratic function by hand, here’s a set of steps that will produce a
reasonably accurate graph:






Use the a value (from any of the three forms) to get the shape:  if a > 0,  if a < 0.
Find the zeros using the easiest method (factoring, square roots, completing the square,
quadratic formula). Provided that there are two zeros, this will give you two points to draw.
Find the vertex (using vertex form or –b/2a or averaging the zeros). This usually gives a
third point to draw.
Find the y-intercept (evaluate f(0), or just take the c value from ax2 + bx + c form). This
usually gives a fourth point to draw.
Draw a dotted line for the axis of symmetry (vertical line through the vertex). Then you can
often get a fifth point by drawing the reflected image of the y-intercept point.
Make sure that all the points you’ve found fit with the shape that you anticipated ( or ).
If everything looks OK, draw a parabola shape passing through the points you have.
Example: Make a graph of f(x) = x2 + 6x + 8.
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




The a value is 1, so expect the  shape.
Factoring gives the zeros: f(x) = (x + 2)(x + 4) so there are points (–2, 0) and (–4, 0).
Completing-the-square leads to vertex form: f(x) = (x + 3)2 – 1 so the vertex is (–3, –1).
f(0) = 8 so the y-intercept is at (0, 8).
The axis of symmetry x = –3 shows that there’s a reflected image point at (–6, 8).
Draw a graph using the five known points:
Name:
Date: ______________
Algebra 2
Quadratic Functions: Putting it all Together page 6
2. Do all of the following for the function f(x) = 2x2 – 8x + 6, without using a calculator.
a. Using factoring, find the zeros.
b. Using completing-the-square, find the vertex.
c. Find the y-intercept.
d. Draw the 4 points found so far, use the
axis of symmetry to find a 5th point, then
sketch the graph of f(x).
Name:
Date: ______________
Algebra 2
Quadratic Functions: Putting it all Together page 7
3. Using the same sequence of steps as in problem 2, find five points then draw the graph for
f(x) = –x2 + 6x – 8, without using a calculator.
Name:
Date: ______________
4. Do all of the following for the function f(x) =
Algebra 2
Quadratic Functions: Putting it all Together page 8
1
2
x2 + 2x – 4, without using a calculator.
a. Using the Quadratic Formula, find the zeros.
b. Using a formula for the x-coordinate of the vertex, find the vertex.
c. Find the y-intercept.
d. Draw the 4 points found so far (some of
them will involve non-whole numbers but
put them in approximately the correct
place). Use the axis of symmetry to find a
5th point, then sketch the graph of f(x).