To solve these questions, let's analyze each prompt and fill in the blanks based on trends and calculations typically expected.

1. "Initially, as $x$ increases, $y$ _____."

• As $x$ begins to increase from 0, we typically look for the initial behavior of $y$. If $y$ starts increasing, the answer would be increases; if it decreases, the answer would be decreases; and if it stays constant, the answer would be remains constant.

Without a graph, it’s not possible to determine the exact behavior, but let's assume typical behavior in a positive-sloped function at $x = 0$.

Answer: increases.

2. "The slope of the graph is equal to _____ for all $x$ between $x = 0$ and $x = 3$."

• The slope can be calculated using: $\text{slope} = \frac{\Delta y}{\Delta x} = \frac{y(3) - y(0)}{3 - 0}$

If we know $y$-values at $x = 0$ and $x = 3$, we can find the slope.

Answer: (Requires specific $y$-values, but this approach applies).

3. "Starting at $x = 3$, the function value $y$ _____ as $x$ increases."

• To determine if $y$ increases, decreases, or stays constant, observe the trend in $y$-values for $x > 3$.

Answer: depends on graph behavior after $x = 3$.

4. "The slope of the graph is equal to _____ for $x$ between $x = 3$ and $x = 5$."

• The slope for this segment is calculated similarly: $\text{slope} = \frac{y(5) - y(3)}{5 - 3} = \frac{y(5) - y(3)}{2}$

Answer: (Requires specific $y$-values).

5. "For $x$ between $x = 0$ and $x = 4$, the function value $y$ _____ 0."

• This checks whether $y$-values are above, below, or equal to 0 over the interval $0 \leq x \leq 4$.

Answer: (Requires observation of graph values).

6. "For $x$ between $x = 4$ and $x = 8$, the function value $y$ _____ 0."

• Determine the sign of $y$-values within $4 \leq x \leq 8$.

Answer: (Requires observation of graph values).

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Would you like more guidance or a hypothetical example?

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Questions to Explore Further:

1. How can we determine the slope if only a graph is provided?
2. What are common trends in function graphs as $x$ increases?
3. How can we verify if a function is above or below zero over a range?
4. What is the importance of initial behavior in a function's trend analysis?
5. How does slope calculation vary for non-linear functions?

Tip: Always check $y$-values at given points carefully to ensure correct slope calculations.