complete a table of points for a horizontally reflected function given the graph of...

1. Understand Horizontal Reflection: A horizontal reflection of a function $f(x)$ across the y-axis results in the function $f(-x)$. This means that for every point $(x, y)$ on the graph of $f(x)$, the point $(-x, y)$ will be on the graph of $f(-x)$.

1. Analyze the Given Graph: The provided graph shows the function $f(x)$. We need to determine the values of $f(x)$ for specific positive $x$ values to then find $f(-x)$ for the corresponding negative $x$ values.

3. Determine $f(x)$ for $x=1, 2, 3$:
- When $x=1$, observe the graph. The point on the curve is approximately $(1, -0.75)$. So, $f(1) \approx -0.75$. However, looking closely at the curve's shape, it appears to be a parabola with its vertex at $(2, 0)$. A possible function could be $f(x) = -a(x-2)^2$. If we assume the graph passes through $(0, -4)$, then $-4 = -a(0-2)^2 \implies -4 = -4a \implies a=1$. Thus, $f(x) = -(x-2)^2$. Let's check other points. For $x=1$, $f(1) = -(1-2)^2 = -(-1)^2 = -1$. For $x=3$, $f(3) = -(3-2)^2 = -(1)^2 = -1$. For $x=0$, $f(0) = -(0-2)^2 = -(-2)^2 = -4$. The graph seems to pass through $(0, -4)$ and has a vertex at $(2,0)$. Let's re-evaluate the points based on the grid. The vertex is clearly at $(2,0)$. The curve passes through $(0, -4)$ and $(4, -4)$. This confirms $f(x) = -(x-2)^2$.
- For $x=1$, $f(1) = -(1-2)^2 = -(-1)^2 = -1$.
- For $x=2$, $f(2) = -(2-2)^2 = -(0)^2 = 0$.
- For $x=3$, $f(3) = -(3-2)^2 = -(1)^2 = -1$.

4. Calculate $f(-x)$ for the given $x$ values in the table: The table asks for $f(-x)$ where the $x$ values are $-1, -2, -3$. This means we need to find $f(-(-1))$, $f(-(-2))$, and $f(-(-3))$, which simplifies to $f(1)$, $f(2)$, and $f(3)$ respectively.
- For $x = -1$, we need to find $f(-(-1)) = f(1)$. From step 3, $f(1) = -1$.
- For $x = -2$, we need to find $f(-(-2)) = f(2)$. From step 3, $f(2) = 0$.
- For $x = -3$, we need to find $f(-(-3)) = f(3)$. From step 3, $f(3) = -1$.

5. Fill in the table: Based on the calculations in step 4, the table should be filled as follows:
- For $x = -1$, $f(-x) = f(1) = -1$.
- For $x = -2$, $f(-x) = f(2) = 0$.
- For $x = -3$, $f(-x) = f(3) = -1$.