Question: A leaf hangs from a branch 12 feet in the air. It falls to the ground at a rate of 0.25 feet per second. Which graph could represent the leaf's height in feet as a function of time, in seconds, after leaving the branch?
1. **State the problem:**
We want to find the function that represents the height $h(t)$ of a leaf falling from a branch 12 feet high, falling at a rate of 0.25 feet per second.
2. **Identify the variables and formula:**
- Initial height: $h(0) = 12$ feet.
- Rate of fall (velocity): $-0.25$ feet per second (negative because height decreases).
The height as a function of time $t$ can be modeled by a linear equation:
$$h(t) = h_0 + vt$$
where $h_0$ is the initial height and $v$ is the velocity.
3. **Apply the values:**
$$h(t) = 12 - 0.25t$$
4. **Find the x-intercept (time when leaf hits the ground):**
Set $h(t) = 0$:
$$0 = 12 - 0.25t$$
$$0.25t = 12$$
$$t = \frac{12}{0.25} = 48$$
So the leaf hits the ground at $t = 48$ seconds.
5. **Analyze the graphs:**
- Graph 1: y-intercept 15, x-intercept 12 (incorrect initial height and time).
- Graph 2: y-intercept 12, x-intercept 20 (incorrect x-intercept, should be 48).
- Graph 3: y-intercept 12, x-intercept beyond 20, shallow negative slope (matches initial height and slope close to $-0.25$).
- Graph 4: y-intercept 20, x-intercept 12 (incorrect initial height).
6. **Conclusion:**
Graph 3 best represents the leaf's height as a function of time because it starts at 12 feet and has a shallow negative slope consistent with $-0.25$ feet per second, with the x-intercept beyond 20 seconds, closer to the calculated 48 seconds.
**Final answer:** Graph 3 represents the leaf's height function.