susans science class is performing an experiment that involves dropping objects from various heights…
susans science class is performing an experiment that involves dropping objects from various heights, starting close to the ground and working upward to 8 feet. the function $t(x)=\frac{1}{4}sqrt{x}$, where $x$ represents the distance from the ground, represents the time it takes for the object susan drops to hit the ground. the graph represents the function $t(x)=\frac{1}{4}sqrt{x}$. identify some of the key features of the graph. that is, determine if the function is monotonically increasing or decreasing, state the end - behavior, find the $x$- and $y$-intercepts, find the maximum or minimum, and state the domain and the range of the graph (without considering the context).
Answer
Explanation:
Step1: Determine monotonicity
The function is $t(x)=\frac{1}{4}\sqrt{x}$. The derivative of $\sqrt{x}$ is $\frac{1}{2\sqrt{x}}$, and the derivative of $t(x)$ is $\frac{1}{8\sqrt{x}}> 0$ for $x > 0$. So the function is monotonically increasing.
Step2: Find end - behavior
As $x\rightarrow+\infty$, $t(x)=\frac{1}{4}\sqrt{x}\rightarrow+\infty$.
Step3: Find x - intercept
Set $t(x) = 0$. Then $\frac{1}{4}\sqrt{x}=0$, which gives $x = 0$. So the x - intercept is $(0,0)$.
Step4: Find y - intercept
Set $x = 0$. Then $t(0)=\frac{1}{4}\sqrt{0}=0$. So the y - intercept is $(0,0)$.
Step5: Find minimum/maximum
Since the function is monotonically increasing, the minimum value occurs at $x = 0$ and $t(0)=0$. There is no maximum.
Step6: Find domain and range
For the square - root function, the domain is $x\geq0$. Since $t(x)=\frac{1}{4}\sqrt{x}\geq0$, the range is $y\geq0$.
Answer:
- Monotonicity: Monotonically increasing.
- End - behavior: As $x\rightarrow+\infty$, $t(x)\rightarrow+\infty$.
- x - intercept: $(0,0)$.
- y - intercept: $(0,0)$.
- Minimum/Maximum: Minimum value of 0 at $x = 0$, no maximum.
- Domain: $x\geq0$.
- Range: $y\geq0$.