Answer:
The graph of f(x) is shifted up 50 units
The graph of f(x) is shifted right 10 units
The graph of f(x) is reflected over the x-axis
Step-by-step explanation:
we have
[tex]f(x)=x^{2}[/tex]
This is a vertical parabola open upward
The vertex is a minimum
The vertex is the origin (0,0)
[tex]g(x)=-5x^{2}+100x-450[/tex]
This is a vertical parabola open downward
The vertex is a maximum
The first thing to note is that fx) is a parabola that opens up and g(x) opens down, so a reflection across the x-axis must have been applied.
Find the vertex of g(x)
Convert to vertex form
[tex]g(x)=-5x^{2}+100x-450[/tex]
Complete the square
[tex]g(x)=-5(x^{2}-20x)-450[/tex]
[tex]g(x)=-5(x^{2}-20x+100)-450+500[/tex]
[tex]g(x)=-5(x^{2}-20x+100)+50[/tex]
[tex]g(x)=-5(x-10)^{2}+50[/tex]
The vertex is the point (10,50)
so
To translate the vertex of (0,0) to (10,50)
The rule of the translation is
(x,y) ------> (x+10,y+50)
That means ----> The translation is 10 units at right and 50 units up
therefore
The transformations are
The graph of f(x) is shifted up 50 units
The graph of f(x) is shifted right 10 units
The graph of f(x) is reflected over the x-axis
