Which phrase best describes the translation from the graph [tex]y=(x-5)^2+7[/tex] to the graph of [tex]y=(x+1)^2-2[/tex] ?
A. 6 units left and 9 units down
B. 6 units right and 9 units down
C. 6 units left and 9 units up
D. 6 units right and 9 units up
Answer (2)
Identify the vertices of the two parabolas: ( 5 , 7 ) and ( − 1 , − 2 ) .
Calculate the horizontal translation: − 1 − 5 = − 6 (6 units left).
Calculate the vertical translation: − 2 − 7 = − 9 (9 units down).
The translation is 6 units left and 9 units down: $\boxed{\text{6 units left and 9 units down}}.
Explanation
Understanding the Problem We are given two parabolas, y = ( x − 5 ) 2 + 7 and y = ( x + 1 ) 2 − 2 . We want to find the translation that maps the first parabola to the second. The key to solving this problem is to recognize that the vertex form of a parabola, y = a ( x − h ) 2 + k , directly reveals the vertex of the parabola, which is located at the point ( h , k ) .
Finding the Vertex of the First Parabola The first parabola, y = ( x − 5 ) 2 + 7 , is in vertex form. Its vertex is at the point ( 5 , 7 ) . This is because the equation is of the form y = ( x − h ) 2 + k , where h = 5 and k = 7 .
Finding the Vertex of the Second Parabola Similarly, the second parabola, y = ( x + 1 ) 2 − 2 , is also in vertex form. We can rewrite it as y = ( x − ( − 1 ) ) 2 + ( − 2 ) . Thus, its vertex is at the point ( − 1 , − 2 ) .
Determining the Translation To find the translation from the first vertex to the second, we need to determine how many units we need to move horizontally and vertically. The horizontal translation is the difference in the x-coordinates: − 1 − 5 = − 6 . This means we need to move 6 units to the left. The vertical translation is the difference in the y-coordinates: − 2 − 7 = − 9 . This means we need to move 9 units down.
Conclusion Therefore, the translation from the graph of y = ( x − 5 ) 2 + 7 to the graph of y = ( x + 1 ) 2 − 2 is 6 units left and 9 units down.
Examples
Understanding translations of graphs is useful in many fields, such as computer graphics, where objects need to be moved around on the screen. For example, if you have a design represented by a function, you can easily reposition it by applying horizontal and vertical translations. This is also applicable in physics, where understanding how the position of an object changes over time can be modeled using translations. For instance, if you know the initial position of a projectile and its trajectory, you can predict its position at any given time by applying the appropriate translations.
The translation from the graph of y = ( x − 5 ) 2 + 7 to y = ( x + 1 ) 2 − 2 involves moving 6 units left and 9 units down. The starting vertex is at ( 5 , 7 ) and the ending vertex is at ( − 1 , − 2 ) . Therefore, the correct answer is A.
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