GUIDED NOTES – Lesson 2-1
Graphing Quadratic Functions
Name: ______________________ Period: ___
Objective: I can graph quadratic functions in standard form, vertex form, and factored form using a table with
applying key points and symmetry.
Quadratic functions are functions where your input/independent variable (
) is raised to the power (
).
This causes the function to behave differently than a linear function. The graph of a quadratic function is
called a ________________. Here are three forms of quadratics that we will learn to graph with today.



f(x) = ax2 + bx + c
f(x) = a(x – h)2 + k
f(x) = a(x – x1)(x – x2)
______________________
______________________
_____________________
Notice that all three forms above start with the ‘a’ term. If that term is
positive (a), the graph opens _____. If that term is negative (-a), the graph
opens down.
All quadratic equations have a ______________, which is the turning point of
the graph. Quadratic graphs are also symmetrical across the axis of
symmetry, which runs through the vertex.
Also notice this particular graphed quadratic function crosses the y-axis
______ and the x-axis _________.
The vertex is:
The y-intercept is:
The axis of symmetry is:
The x-intercepts are:
There is a short and sweet overview of the general graph of a quadratic function. Using these key points, graphing can
be done more fluently in every form. Not all of these points will be applicable to every form necessarily.
Graphing in STANDARD FORM – First we need to pull out the vertex from the function.
We do that with the vertex formula, which tells us the x-value of the vertex.
Vertex formula:
You then substitute the x-value you get into the original function to get the y-value.
Lastly, pick x-values above and below the vertex to plug in and get more
points.
EXAMPLE - Graph the function: f(x) = 4x2 – 8x + 1
x
f(x)
Graphing VERTEX FORM – When a quadratic function is in vertex form, the vertex is easy to pull out.
f(x) = a(x – h)2 + k
EXAMPLES: f(x) = 2(x – 2)2 + 4
The vertex is always the values of (h, k)
Vertex:
f(x) = -4(x + 3)2 – 5
Vertex:
To graph in vertex form, put the vertex in the table and pick x values above and below to get more points.
EXAMPLE - Graph the function: f(x) = 2(x – 3)2 – 4
x
f(x)
Graphing in FACTORED FORM – In this form, the equation has been factored out. This time the x-intercepts
(aka zeros) are easy to see. Figure out what value of x will give you 0 for f(x)/y. Plot these points and put
them in your table. The vertex has an x-value between your two x-intercepts (because of symmetry), so that is
𝑥 +𝑥
another x-point to plug in and get the y-value of the vertex. The formal calculation is 1 2 2.
EXAMPLE - Graph the function: f(x) = (x – 6)(x – 2)
x
f(x)
EXAMPLE - Graph the function: f(x) = (x + 1)(x – 4)
x
f(x)