Question: Use the transformed function $g(x)$ in each graph or table to identify the parent function, find the value of $k$, $h$, and/or $a$, and then write transformed function.
29.
f(x): U-shape opening up, vertex at $(0, 0)$, narrow parabola, standard upward opening; position_hint = top-left
g(x): U-shape opening down, vertex at $(4, 0)$, same width as $f(x)$, reflected over the x-axis and shifted right 4; position_hint = bottom-right
1. **State the problem:**
We have a parent function $f(x)$ which is a narrow parabola opening upwards with vertex at $(0,0)$.
The transformed function $g(x)$ is a parabola opening downwards, vertex at $(4,0)$, same width as $f(x)$, reflected over the x-axis and shifted right by 4.
2. **Identify the parent function:**
The parent function for a narrow parabola opening upwards is typically:
$$f(x) = x^2$$
Since it is narrow, it might be $f(x) = ax^2$ with $a > 1$, but since $g(x)$ has the same width, we assume $a=1$ for simplicity.
3. **Find the transformations:**
- Reflection over the x-axis means multiplying by $-1$.
- Shift right by 4 means replacing $x$ by $(x - 4)$.
So the transformed function is:
$$g(x) = -f(x - 4) = -(x - 4)^2$$
4. **Write the transformed function:**
$$g(x) = -(x - 4)^2$$
5. **Summary:**
- Parent function: $f(x) = x^2$
- $a = -1$ (reflection over x-axis)
- $h = 4$ (shift right 4)
- $k = 0$ (no vertical shift)
Hence, the transformed function is:
$$g(x) = - (x - 4)^2$$