Question

the water level in a bathtub after a certain amount of time is represented by the function w(x) graphed to the right. what is the rate of change between 0 and 3 minutes? what does y - intercept mean in context to the problem? state the domain and range for w(x). where is the graph decreasing? what does this mean in the context of the situation? evaluate w(8). what does this answer mean? evaluate w(x)=8. what does this answer mean?

Explanation:

Step1: Recall rate - of - change formula

The rate of change of a function $y = f(x)$ between $x_1$ and $x_2$ is $\frac{f(x_2)-f(x_1)}{x_2 - x_1}$. For the function $w(x)$ between $x_1 = 0$ and $x_2=3$, we need to find the values of $w(0)$ and $w(3)$ from the graph.

Step2: Determine domain

The domain of a function is the set of all possible input values (x - values). Looking at the x - axis of the graph of $w(x)$, we find the minimum and maximum x - values.

Step3: Determine range

The range of a function is the set of all possible output values (y - values). Looking at the y - axis of the graph of $w(x)$, we find the minimum and maximum y - values.

Step4: Evaluate $w(8)$

To evaluate $w(8)$, we find the y - value on the graph when $x = 8$.

Step5: Solve $w(x)=8$

To solve $w(x)=8$, we find the x - value(s) on the graph when $y = 8$.

Since the graph is not provided with numerical values, we'll assume general steps for interpretation:

What is the rate of change between 0 and 3 minutes?

Let $w(0)=y_1$ and $w(3)=y_2$. The rate of change $r=\frac{w(3)-w(0)}{3 - 0}=\frac{y_2 - y_1}{3}$.

What does y - intercept mean in context to the problem?

The y - intercept is the value of $w(x)$ when $x = 0$. It represents the initial water level in the bathtub.

State the domain and range for $w(x)$

Domain: The set of all x - values for which the function $w(x)$ is defined. If the graph starts at $x = a$ and ends at $x = b$, the domain is $[a,b]$.
Range: The set of all y - values that the function $w(x)$ takes. If the minimum y - value is $m$ and the maximum is $M$, the range is $[m,M]$.

Where is the graph decreasing? What does this mean in the context of the situation?

The graph is decreasing when the slope of the function is negative. In the context of the water - level problem, it means the water level in the bathtub is decreasing.

Evaluate $w(8)$. What does this answer mean?

We find the y - value on the graph when $x = 8$. It represents the water level in the bathtub 8 minutes after the start.

Evaluate $w(x)=8$. What does this answer mean?

We find the x - value(s) on the graph when $y = 8$. It represents the time(s) when the water level in the bathtub is 8 (units of water - level measurement).

Answer:

• Rate of change between 0 and 3 minutes: $\frac{w(3)-w(0)}{3}$
• Y - intercept: Initial water level in the bathtub
• Domain: Set of all x - values for which $w(x)$ is defined
• Range: Set of all y - values that $w(x)$ takes
• Graph is decreasing: When slope is negative; water level is decreasing
• $w(8)$: Water level 8 minutes after start
• $w(x)=8$: Time(s) when water level is 8