Answer
a) $f(x)=x^2$
b) Vertical stretch, reflection across the $x$-axis, vertical shift
c) See graph
d) $g(x)=-2f(x)+1$
Work Step by Step
We are given the function:
$g(x)=-2x^2+1$
a) The parent function is:
$f(x)=x^2$
b) Vertically stretch $f(x)$ by a factor of 2 to get $a(x)=2x^2$.
Reflect $a(x)$ across the $x$-axis to get $b(x)=-2x^2$.
Vertically shift $b(x)$ one unit upward to get $g(x)=-2x^2+1$.
c) Graph the transformations.
d) Use function notation to write $g$ in terms of $f$:
$g(x)=-2f(x)+1$
