f(x)=x^2; vertical shrink by a factor of 1/2 and a reflection in the y-axis, followed by a translation 1 unit down
The Correct Answer and Explanation is:
To transform the function f(x)=x2f(x) = x^2f(x)=x2 as described, we need to apply each transformation step by step. Let’s break it down:
- Vertical Shrink by a Factor of 12\frac{1}{2}21:
A vertical shrink by a factor of 12\frac{1}{2}21 means that the output values (the yyy-values) of the function are halved. This can be achieved by multiplying the function by 12\frac{1}{2}21. The function becomes: f(x)=12x2f(x) = \frac{1}{2} x^2f(x)=21x2 This transformation reduces the height of the graph by half. - Reflection in the y-axis:
A reflection across the yyy-axis involves changing the sign of the xxx-variable. Mathematically, this can be represented by replacing xxx with −x-x−x in the function. The function becomes: f(x)=12(−x)2f(x) = \frac{1}{2} (-x)^2f(x)=21(−x)2 Since squaring −x-x−x still results in x2x^2×2, the function simplifies to: f(x)=12x2f(x) = \frac{1}{2} x^2f(x)=21x2 This step doesn’t affect the graph of the function because squaring xxx or −x-x−x produces the same result. Hence, the graph is still symmetric about the yyy-axis. - Translation 1 Unit Down:
A translation of 1 unit down means we subtract 1 from the function. The new function becomes: f(x)=12×2−1f(x) = \frac{1}{2} x^2 – 1f(x)=21x2−1 This shifts the entire graph of the function downward by 1 unit.
Final Function:
After applying all these transformations, the final function is:f(x)=12×2−1f(x) = \frac{1}{2} x^2 – 1f(x)=21x2−1
Explanation:
The transformations applied to the original function f(x)=x2f(x) = x^2f(x)=x2 are straightforward:
- The vertical shrink reduces the amplitude of the parabola, making it “flatter.”
- The reflection across the yyy-axis doesn’t change the shape since the original function is already symmetric about the yyy-axis.
- The downward translation shifts the entire graph 1 unit lower, maintaining the shape but adjusting the position on the coordinate plane.
Thus, the overall transformation results in a parabola that is narrower than the original (due to the vertical shrink), symmetric about the yyy-axis (since the reflection doesn’t change the graph), and shifted down by 1 unit.
