Quadratic Functions:
We have learned three forms of quadratic functions
-Vertex Form:
-Factored form:
-Standard Form:
When given a graph like the one below we can write the equation for that quadratic
in all three forms.
You can start with any form but for this example we will start with vertex form.
First identify the vertex. This is the point where the parabola turns around. In this
example the vertex is located at (2,4). This tells us that h=2 and k=4 so we plug
these values into the vertex form as shown below:
The next step is to solve for a. To do this choose any other point on the parabola
and plug in for x and y. I am going to choose the point (1,0) for this example which
mean I will plug 1 in for x and 0 in for y.
Now that we know a we can plug everything into vertex form.
Final Answer for vertex form!
Now let’s find factored form!
The terms and are the x values of the root, so we must identify the roots.
Looking at the graph we can see the roots (or x-intercepts) are the points (1,0) and
(3,0). This tells us that
and
Since this is the same parabola, the a value has not changed so a=-4 and now we
just plug every thing in.
Final answer for factored form!
Now we can use either vertex form or factored form and expand it into standard
form. I will show both here but you only need to do one.
Vertex form to standard form:
Factored form to Standard form:
Distribute using F.O.I.L.
Distribute using F.O.I.L.
Combine like terms
Combine like terms
Distribute the -4.
Distribute the -4.
Combine like terms.
Final answer for standard form!
Now we have found all three forms of the equation for the graph!
Vertex form:
Factored form:
Standard form:
NOW YOU TRY! Use the graph on the next page and see if you can find all 3
forms!
Scroll to the next page for answers. Come see me if you have questions
Vertex form:
Factored form:
Standard form: