Functions Worksheet - Math 121: Extra Practice For Test 2 Page 3
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2
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53
Polynomial Functions
Recall that a polynomial of degree n is a function of the form
1
P (x) = a x + a
x
+ ... + a
x + a
,
1
1
0
where a , a
, ..., a
, a
are real numbers and n is a nonnegative integer. The coefficient in front of
1
1
0
x , a , is called a leading coefficient.
(a) What is the domain of any polynomial function?
(b) What is a zero of a polynomial?
(c) Describe what can happen with the graph of the function at the real zeros.
(d) What can you say about the multiplicity of a zero if the graph of the polynomial crosses the
x axis at that zero?
(e) What do you understand by ’end behavior’ ?
(f) What is a turning point?
(g) How many turning points can a polynomial of degree n have?
(h) If f (x) < 0 in some interval, what can you say about the graph of f (x) in that interval?
TO SKETCH THE GRAPH OF A POLYNOMIAL FUNCTION:
(a) Determine the degree of the polynomial.
(b) Determine the sign of the leading coefficient.
(c) Based on the degree and the sign of the leading coefficient, determine the end behavior of the
graph.
(d) Make sure that the function is in completely factored form.
(e) Find the zeros.
(f) Find the multiplicities, m, of each of the zeros.
(g) Depending whether m is odd/even, determine if the graph will cross/touch the x-axis at each
real zero.
(h) Find the y-intercept.
(i) Determine the maximum number of turning points the graph can have.
(j) If you need additional information in order to plot the graph: plot the zeros/x-intercepts on
a number line and pick a point in each interval to determine if the graph of the function is
above/below the x axis in that interval.
(k) Plot the information obtained in the above steps and use a smooth, continuous curve to plot the
graph.
11. Follow the steps given above to sketch the graph of each of the given polynomial functions:
2
2
(a) f (x) = (x
4)(x + 2)
(x
2)
(d) h(x) = (x
25)(x + 5)
4
2
(e) q(x) =
2x
+ 4x
3
(b) g(x) =
4x
+ 4x
1
2
2
2
(f) r(x) =
(x
+ 1)(x
3)
(c) p(x) = x(x
1)
(x + 3)(x + 1)
2
12. Form a polynomial with zeros at -2, 0, and 7 and multiplicities 2, 3, 1 respectively, whose leading
coefficient -3. What is the degree of the resulting polynomial?
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