Question

given the polynomial function ( f(x) = (x + 3)^2(x - 7)^2 ), using parts (a) through (e).

(a) determine the end behavior of the graph of the function.
the graph of ( f ) behaves like ( y = square ) for large values of ( |x| ).

(b) find the ( x )- and ( y )-intercepts of the graph of the function.
the ( x )-intercept(s) is/are ( square )
(simplify your answer. type an integer or a fraction. use a comma to separate answers as needed. type each answer only once.)
the ( y )-intercept is ( square )
(simplify your answer. type an integer or a fraction.)

(c) determine the zeros of the function and their multiplicity. use this information to determine whether the graph crosses or touches the ( x )-axis at each ( x )-intercept.
the zero(s) of ( f ) is/are ( square )
(simplify your answer. type an integer or a fraction. use a comma to separate answers as needed. type each answer only once.)
the lesser zero is a zero of multiplicity ( square ), so the graph of ( f ) (\boldsymbol{
abla}) the ( x )-axis at ( x = square ). the greater zero is a zero of multiplicity ( square ), so the graph of ( f ) (\boldsymbol{
abla}) the ( x )-axis at ( x = square ).

(d) determine the maximum number of turning points on the graph of the function.
( square )
(type a whole number.)

(e) use the information to draw a complete graph of the function. choose the correct graph.

a.
graph a

b.
graph b

c.
graph c

d.
graph d

Explanation:

Answer:

(a) $y = x^3$
(b) x-intercepts: $-3, 7$; y-intercept: $63$
(c) zeros: $-3, 7$; lesser zero $-3$ has multiplicity $2$, touches the x-axis at $x = -3$; greater zero $7$ has multiplicity $1$, crosses the x-axis at $x = 7$
(d) $2$
(e) A. [Graph showing a touch at $x = -3$ and a cross at $x = 7$]