Question

which transformations have been applied to the graph of ( f(x) = x^2 ) to produce the graph of ( g(x) = -5x^2 + 100x - 450 )? select three options.

• the graph of ( f(x) = x^2 ) is shifted down 50 units.
• the graph of ( f(x) = x^2 ) is shifted up 50 units.
• the graph of ( f(x) = x^2 ) is shifted left 10 units.
• the graph of ( f(x) = x^2 ) is shifted right 10 units.
• the graph of ( f(x) = x^2 ) is reflected over the x - axis.

Explanation:

Step1: Rewrite $g(x)$ in vertex form

Complete the square for $g(x)=-5x^2+100x-450$:
Factor out -5 from first two terms:
$g(x) = -5(x^2 - 20x) - 450$
Add and subtract $(\frac{-20}{2})^2=100$ inside the parentheses:
$g(x) = -5(x^2 - 20x + 100 - 100) - 450$
Rewrite as:
$g(x) = -5((x-10)^2 - 100) - 450$
Expand:
$g(x) = -5(x-10)^2 + 500 - 450$
Simplify:
$g(x) = -5(x-10)^2 - 50$

Step2: Compare to $f(x)=x^2$

Transformation rules for $a(x-h)^2+k$ from $x^2$:

• $a=-5$: negative $a$ means reflection over x-axis.
• $h=10$: positive $h$ means shift right 10 units.
• $k=-50$: negative $k$ means shift down 50 units.

Answer:

1. The graph of $f(x) = x^2$ is shifted down 50 units.
2. The graph of $f(x) = x^2$ is shifted right 10 units.
3. The graph of $f(x) = x^2$ is reflected over the x-axis.