**Initially, as x*x* increases, y*y* _____ .**

Determine whether y*y* increases, decreases, or remains constant as x*x* begins to increase from 0. Look at the general trend at the beginning of the graph.

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

The slope can be calculated by finding the change in y*y* over the change in x*x* in this range. Check the values of y*y* at x=0*x*=0 and x=3*x*=3 and use slope=ΔyΔxslope=Δ*x*Δ*y*​.

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

Check whether the function y*y* rises, falls, or stays constant for x*x* values greater than 3.

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

Similar to step 2, find the slope in this specific range by looking at y*y*-values at x=3*x*=3 and x=5*x*=5.

**For x*x* between x=0*x*=0 and x=4*x*=4, the function value y*y* _____ 0.**

Check if y*y*-values are above, below, or equal to 0 within this range.

**For x*x* between x=4*x*=4 and x=8*x*=8, the function value y*y* _____ 0.**  i need the answer

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.

Initially, as x*x* increases, y*y* _____. Determine whether y*y* increases, decreases, or remains constant as x*x* begins to increase from 0. Look at the general trend at the beginning of the graph. The slope of the graph is equal to _____ for all x*x* between x=0*x*=0 and x=3*x*=3. The slope can be calculated by finding the change in y*y* over the change in x*x* in this range. Check the values of y*y* at x=0*x*=0 and x=3*x*=3 and use slope=ΔyΔxslope=Δ*x*Δ*y*​. Starting at x=3*x*=3, the function value y*y* _____ as x*x* increases. Check whether the function y*y* rises, falls, or stays constant for x*x* values greater than 3. The slope of the graph is equal to _____ for x*x* between x=3*x*=3 and x=5*x*=5. Similar to step 2, find the slope in this specific range by looking at y*y*-values at x=3*x*=3 and x=5*x*=5. For x*x* between x=0*x*=0 and x=4*x*=4, the function value y*y* _____ 0. Check if y*y*-values are above, below, or equal to 0 within this range. For x*x* between x=4*x*=4 and x=8*x*=8, the function value y*y* _____ 0.

Learn how to determine function behavior and calculate slope in a graph. This exercise involves analyzing y-values as x increases, calculating slope between specified intervals, and understanding function trends. Ideal for Grades 9-10 students looking to strengthen skills in graph analysis and slope calculations.

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