Solve the problem.
-The accompanying figure shows the graph of $y = - x ^ { 2 }$ shifted to a new position. Write the equation for the new graph.

A) $y = - ( x + 2 ) ^ { 2 } + 3$
B) $y = - ( x - 2 ) ^ { 2 } + 3$
C) $y = - ( x - 2 ) ^ { 2 } - 3$
D) $y = - ( x + 3 ) ^ { 2 } + 2$

Question

Solve the problem.
-The accompanying figure shows the graph of $y = - x ^ { 2 }$ shifted to a new position. Write the equation for the new graph.

A) $y = - ( x + 2 ) ^ { 2 } + 3$ 
B) $y = - ( x - 2 ) ^ { 2 } + 3$ 
C) $y = - ( x - 2 ) ^ { 2 } - 3$ 
D) $y = - ( x + 3 ) ^ { 2 } + 2$

Quizplus · Accepted Answer

The vertex of the original graph is at (0,0), which means the equation is written in the form $y=a(x-0)^2+0$ or simply $y=ax^2$. The negative coefficient of x squared means that the graph is downward facing. 
The new graph appears to be shifted 2 units to the right and 3 units upward. The vertex of the new graph is therefore at (-2,3). We want the equation to have this vertex and still be downward facing.

If we consider the vertex form of a quadratic equation, $y=a(x-h)^2+k$, we can see that the vertex is at (h,k). So we want to shift the x values right by 2 units which means we need to use (x-(-2)) or (x+2) in the equation. We also want the graph to go downward, so we need to use a negative value of a.

Putting it all together, we get $y=-a(x+2)^2+3$ which matches answer choice B.

Solve the Problem y=−x2y = - x ^ { 2 }y=−x2

Solve the Problem $y = - x ^ { 2 }$