Question

10 which of the following functions has a graph with the vertex and the axis of symmetry to the left of the vertex and the axis of symmetry of the graph of $f(x)=(x - 1)^2 + 1$? select all that apply.
a $g(x)=2(x + 1)^2 + 2$
b $g(x)=2(x - 1)^2 + 2$
c $g(x)=-(x + 1)^2 - 2$
d $g(x)=-(x + 2)^2 + 2$
e $g(x)=2(x - 1)^2 - 1$
f $g(x)=-(x - 2)^2 + 1$

Explanation:

Step1: Identify vertex of $f(x)$

For vertex form $f(x)=a(x-h)^2+k$, vertex is $(h,k)$.
For $f(x)=(x-1)^2+1$, vertex is $(1, 1)$, axis of symmetry $x=1$.

Step2: Define "left of $x=1$"

A vertex has $h < 1$ (axis of symmetry $x=h < 1$).

Step3: Check each function's $h$

• A: $g(x)=2(x+1)^2+2=2(x-(-1))^2+2$, $h=-1 < 1$
• B: $g(x)=2(x-1)^2+2$, $h=1$ (not left)
• C: $g(x)=-(x+1)^2-2=-(x-(-1))^2-2$, $h=-1 < 1$
• D: $g(x)=-(x+2)^2+2=-(x-(-2))^2+2$, $h=-2 < 1$
• E: $g(x)=2(x-1)^2-1$, $h=1$ (not left)
• F: $g(x)=-(x-2)^2+1$, $h=2 > 1$ (not left)

Answer:

A. $g(x)=2(x+1)^2+2$, C. $g(x)=-(x+1)^2-2$, D. $g(x)=-(x+2)^2+2$