the function f(x) = (x+3)^2 is reflected over the x-axis, then how does the reflect...
Check the final answer first, then review the worked steps.
Problem
the function f(x) = (x+3)^2 is reflected over the x-axis, then how does the reflection affect the point (2, 25)?
Answer
\((2, -25)\)
Step-by-step solution
- Understand the transformation: The problem states that the function $f(x) = (x+3)^2$ is reflected over the x-axis. A reflection over the x-axis means that for any point $(x, y)$ on the original graph, the corresponding point on the reflected graph will be $(x, -y)$. The x-coordinate remains the same, and the y-coordinate is negated.
2. Evaluate the original function at the given x-coordinate: We are given a point $(2, 25)$. We need to determine if this point lies on the graph of $f(x) = (x+3)^2$. Let's substitute $x=2$ into the function:
$$f(2) = (2+3)^2 = (5)^2 = 25$$So, the point $(2, 25)$ is indeed on the graph of the original function.
- Apply the reflection rule: Since the reflection is over the x-axis, the x-coordinate of the point remains unchanged, and the y-coordinate is multiplied by -1. The original point is $(2, 25)$.
- Determine the new coordinates: Applying the reflection rule, the new x-coordinate is $2$, and the new y-coordinate is $-25$.
Therefore, the point $(2, 25)$ becomes $(2, -25)$ after reflection over the x-axis.