Answer:
The graph of parent function reflected across x-axis, vertically stretched by factor 2 and shift 5 units left 3 units down.
Step-by-step explanation:
The parent function is
[tex]f(x)=x^2[/tex]
The given function is
[tex]g(x)=-2x^2-20x-53[/tex]
[tex]g(x)=-2(x^2+10x)-53[/tex]
If an expression is [tex]x^2+bx[/tex], then we have to add [tex](\frac{b}{2})^2[/tex] in it to make it perfect square.
In the parenthesis the value of b is 10.
[tex](\frac{10}{2})^2=5^2=25[/tex]
Add and subtract 25 in the parenthesis.
[tex]g(x)=-2(x^2+10x+25-25)-53[/tex]
[tex]g(x)=-2(x^2+10x+25)-2(-25)-53[/tex]
[tex]g(x)=-2(x+5)^2+50-53[/tex]
[tex]g(x)=-2(x+5)^2-3[/tex]
[tex]g(x)=-2f(x+5)-3[/tex] .... (1)
The translation is defined as
[tex]g(x)=kf(x+a)+b[/tex] .... (2)
Where, k is stretch factor, a is horizontal shift and b is vertical shift.
If 0<|k|<1, then the graph compressed vertically by factor k and if |k|>1, then the graph stretch vertically by factor k.
Negative k represents the reflection across x-axis.
If a>0, then the graph shifts a units left and if a<0, then the graph shifts a units right.
If b>0, then the graph shifts b units up and if b<0, then the graph shifts b units down.
From (1) and (2) it is clear that k=-2, a=5 and b=-3.
It means graph of parent function reflected across x-axis, vertically stretched by factor 2 and shift 5 units left, 3 units down.
