Question

For each statement about the graph of the function
y
=
x
3
+
1
, determine whether the statement is true or false.



Statement

Row Headers
True

False

The graph has a
y
-intercept at
y
=
1
.

The graph has a y-intercept at y=1. and True The graph has a y-intercept at y=1. and False
The graph has
x
-intercepts at
x
=
−
1
,
x
=
0
, and
x
=
1
.

The graph has x-intercepts at x=-1, x=0, and x=1. and True The graph has x-intercepts at x=-1, x=0, and x=1. and False
When
x
is an extreme negative value,
y
is an extreme positive value.

When x is an extreme negative value, y is an extreme positive value. and True

Answer 1

The graph is y = x^3 + 1.

1) The graph has a y-intercept at y = 1. — True.
Reason: at x = 0, y = 0^3 + 1 = 1, so the y-intercept is (0,1).

2) The graph has x-intercepts at x = −1, x = 0, and x = 1. — False.
Reason: x-intercepts satisfy x^3 + 1 = 0 ⇒ x = −1 only. At x = 0 and x = 1, y = 1 and y = 2, respectively, not 0.

3) When x is an extreme negative value, y is an extreme positive value. — False.
Reason: as x → −∞, x^3 + 1 → −∞, so y is extremely negative, not positive.