How is the graph of $y=(x-1)^2-3$ transformed to produce the graph of $y=\frac{1}{2}(x+4)^2$?
A. The graph is translated left 5 units, compressed vertically by a factor of $\frac{1}{2}$, and translated up 3 units.
B. The graph is stretched vertically by a factor of $\frac{1}{2}$, translated left 5 units, and translated up 3 units.
C. The graph is translated left 5 units, compressed horizontally by a factor of $\frac{1}{2}$, and translated down 3 units.
D. The graph is stretched horizontally by a factor of $\frac{1}{2}$, translated left 5 units, and translated down 3 units.
Answer (2)
The vertex of the original function y = ( x − 1 ) 2 − 3 is at ( 1 , − 3 ) .
The vertex of the transformed function y = 2 1 ( x + 4 ) 2 is at ( − 4 , 0 ) .
The graph is translated 5 units to the left and 3 units up.
The graph is compressed vertically by a factor of 2 1 . The answer is: The graph is translated left 5 units, compressed vertically by a factor of 2 1 , and translated up 3 units.
Explanation
Understanding the Problem We are given two quadratic functions, y = ( x − 1 ) 2 − 3 and y = 2 1 ( x + 4 ) 2 , and we want to describe the transformations that map the graph of the first function to the graph of the second function.
Analyzing the Original Function The first function is in vertex form, y = ( x − 1 ) 2 − 3 . This tells us that the vertex of the parabola is at the point ( 1 , − 3 ) . The coefficient of the squared term is 1, which means the parabola opens upwards and has a standard width.
Analyzing the Transformed Function The second function is y = 2 1 ( x + 4 ) 2 . This is also in vertex form, and we can see that the vertex of this parabola is at the point ( − 4 , 0 ) . The coefficient of the squared term is 2 1 , which means the parabola opens upwards and is vertically compressed by a factor of 2 1 .
Determining Horizontal Translation To transform the first graph to the second, we need to consider the changes in the vertex and the shape of the parabola. The x-coordinate of the vertex changes from 1 to -4, which means we need to translate the graph horizontally by − 4 − 1 = − 5 units. This is a translation of 5 units to the left.
Determining Vertical Translation The y-coordinate of the vertex changes from -3 to 0, which means we need to translate the graph vertically by 0 − ( − 3 ) = 3 units. This is a translation of 3 units up.
Determining Vertical Compression The coefficient of the squared term changes from 1 to 2 1 . This means we need to vertically compress the graph by a factor of 2 1 .
Summary of Transformations In summary, the graph of y = ( x − 1 ) 2 − 3 is transformed to the graph of y = 2 1 ( x + 4 ) 2 by translating it 5 units to the left, compressing it vertically by a factor of 2 1 , and translating it 3 units up.
Final Answer Therefore, the correct answer is: The graph is translated left 5 units, compressed vertically by a factor of 2 1 , and translated up 3 units.
Examples
Understanding transformations of functions is crucial in many fields. For example, in physics, understanding how graphs of motion change with different initial conditions or forces can be modeled using transformations. Similarly, in economics, transformations can be used to model how supply and demand curves shift due to changes in market conditions. By understanding these transformations, we can make predictions and understand the behavior of various systems.
The graph of y = ( x − 1 ) 2 − 3 is transformed into y = 2 1 ( x + 4 ) 2 by translating it left 5 units, compressing it vertically by a factor of 2 1 , and translating it up 3 units. Therefore, the correct answer is A.
;
