which of the following best describes how the graph of f can be obtained from the graph of y=1/x^2? f(x)=14/x^2
choose the correct answer
a.vertical stretch by a factor of -4
b. reflection over the y-axis, vertical shrink by a factor of 4
c. reflection over the x-axis, stretch by a factor of 4
d. horizontal shift of 4 units to the left
To obtain the graph of f(x) = 14/x^2 from the graph of y = 1/x^2, we need to understand the transformations involved.
Starting with y = 1/x^2, we can see that it is a basic reciprocal function. The graph of y = 1/x^2 is a hyperbola with the vertex at the origin (0, 0) and branches in the first and third quadrants.
Now, let's analyze the given function f(x) = 14/x^2.
1. Vertical Stretch/Shrink: The coefficient in front of the function determines the vertical stretch or shrink. In this case, the coefficient is 14. Since 14 is greater than 1, this means there is a vertical stretch.
2. Horizontal Transformation: The presence of a coefficient (e.g., a constant) in the denominator of the function affects the horizontal transformation. In this case, there is no coefficient in the denominator, so there is no horizontal transformation.
3. Reflections: A negative coefficient in front of the function causes a reflection over the x-axis, while a negative coefficient inside the function causes a reflection over the y-axis.
Now let's evaluate the possible answer choices:
a. Vertical Stretch by a Factor of -4
Since the coefficient is positive, there is a vertical stretch, not a vertical compression. Therefore, option a is incorrect.
b. Reflection over the y-axis, Vertical Shrink by a Factor of 4
Since the coefficient is positive, there is no reflection over the y-axis. Additionally, the presence of the coefficient 4 indicates a vertical stretch, not a vertical shrink. Therefore, option b is incorrect.
c. Reflection over the x-axis, Stretch by a Factor of 4
There is no reflection over the x-axis in the given function. Furthermore, the coefficient 4 indicates a vertical stretch, not a reflection. Therefore, option c is incorrect.
d. Horizontal Shift of 4 Units to the Left
There is no horizontal shift indicated in the given function. Therefore, option d is also incorrect.
In conclusion, none of the given options accurately describe the transformation needed to obtain the graph of f(x) = 14/x^2 from the graph of y = 1/x^2.