Which phrase best describes the translation from the graph $y=(x+2)^2$ to the graph of $y=x^2+3$?
A. 2 units left and 3 units up
B. 2 units left and 3 units down
C. 2 units right and 3 units up
D. 2 units right and 3 units down
Answer (1)
The vertex of y = ( x + 2 ) 2 is ( − 2 , 0 ) .
The vertex of y = x 2 + 3 is ( 0 , 3 ) .
To go from ( − 2 , 0 ) to ( 0 , 3 ) , we move 2 units to the right and 3 units up.
The translation is 2 units right and 3 units up, so the answer is 2 units right and 3 units up .
Explanation
Understanding the Problem We are asked to describe the translation from the graph of y = ( x + 2 ) 2 to the graph of y = x 2 + 3 . To do this, we can compare the vertices of the two parabolas.
Finding the Vertex of the First Parabola The graph of y = ( x + 2 ) 2 is a parabola with vertex at ( − 2 , 0 ) . This is because the square function ( x + 2 ) 2 is minimized when x + 2 = 0 , which means x = − 2 , and the minimum value is 0.
Finding the Vertex of the Second Parabola The graph of y = x 2 + 3 is a parabola with vertex at ( 0 , 3 ) . This is because the square function x 2 is minimized when x = 0 , and the minimum value is 3.
Describing the Translation To go from the vertex ( − 2 , 0 ) to the vertex ( 0 , 3 ) , we need to move 2 units to the right (from x = − 2 to x = 0 ) and 3 units up (from y = 0 to y = 3 ).
Final Answer Therefore, the translation from the graph of y = ( x + 2 ) 2 to the graph of y = x 2 + 3 is 2 units to the right and 3 units up.
Examples
Understanding translations of graphs is crucial in various fields. For instance, in physics, describing the motion of an object often involves translating its position over time. Similarly, in computer graphics, translating objects on the screen is a fundamental operation. Knowing how to describe these translations mathematically allows us to predict and control the behavior of systems in these fields. For example, if you are designing a game and want to move a character from point A to point B, you need to understand the translation required to achieve that movement. If point A is at (-2, 0) and point B is at (0, 3), the character needs to move 2 units to the right and 3 units up.
