2.3 Graphing
Polynomial Functions
Objective:
-graph polynomial functions
-describe the end behavior
Recall from yesterday
What cubic functions look like…
Must have 3 zeros! (real or imaginary)
Will cross x-axis 3 times (usually)
What quartic functions look like…
Must have 4 zeros! (real or imaginary)
Will cross x-axis 4 times (usually)
Sketching the Graph of a
Polynomial
What I expect:
• The zeros of the polynomial
• Appropriate end behavior
Sketch the Graph of a Factored
Polynomial
• Sketch the graph of 𝑓 𝑥 = (𝑥 + 1)(𝑥 − 1)(𝑥 − 2)
– Find and plot zeros
– Determine the end behavior
– Graph it!
Sketch a graph:
f(x)= (x-3)(x+1)(2x-1)
Sketch a graph:
f(x)=(x+3)(x+1)(x-1)(x-2)
Sketch a graph:
f(x) = -x(x+2)(x-1)
Sketch a graph:
f(x) = x(x+3)(x-4)(1-x)
Sketch a graph:
f(x) = (x-1)(x-3)2(x-4)
Uh oh! We
have 2 of the
same root. What
happens now?
The double root
is tangent to the
x-axis (the curve
“bounces off” the
x-axis).
Analyze the effect of a repeated
factor (multiplicity)
• Summary:
– Even multiplicity: then the graph is tangent to the
x-axis at the zero.
– Odd multiplicity: then the graph crosses the xaxis at the zero.
Sketch a graph:
f(x) = (-x-1)(x-3)2
Sketch a graph:
f(x) = x4(x − 2)3(x + 1)2
Now, we aren’t just given the roots. How
can we sketch a graph from this?
𝑓 𝑥 = 𝑥 3 − 4𝑥
Practice Sess
• Source: Textbook
• Page: 66
• Problems: #1-9 (odds), #13, #15
Homework
• In your textbook complete #2-16 (evens)
on pg. 66