Question

1. $f(x) = -x^3 + 1$

rewrite with a, h, k:
$f(x) = \underline{\quad\quad\quad\quad\quad\quad\quad}$
(grid graph here)

a	-1
h	0
k	1

domain (int): $\underline{( -\infty , +\infty )}$
range (int): $\underline{( -\infty , +\infty )}$
inflection point: $( 0 , 1 )$
zero: $( 1 , 0 )$ y-intercept: $( 0 , 1 )$
transformations: $\underline{reflection, up 1}$
end behavior: as $x \to -\infty$, $y \to \underline{ -\infty }$
as $x \to +\infty$, $y \to \underline{ +\infty }$

Explanation:

Step1: Recall the cubic function form

The general form of a cubic function with transformations is \( f(x) = a(x - h)^3 + k \), where \( a \) is the vertical stretch/compression and reflection factor, \( h \) is the horizontal shift, and \( k \) is the vertical shift.

Step2: Substitute \( a \), \( h \), \( k \) values

Given \( a = -1 \), \( h = 0 \), and \( k = 1 \), substitute these into the general form:
\( f(x) = -1(x - 0)^3 + 1 \)
Simplify the expression:
\( f(x) = -x^3 + 1 \) (which matches the original function, confirming the substitution is correct)

Answer:

\( f(x) = -1(x - 0)^3 + 1 \) (or simplified as \( f(x) = -x^3 + 1 \))