Question

is the table Linear or nonlinear?
x -2 -1 0 1 2
y -8 -3 0 1 0

Answer 1

is the table Linear or nonlinear?
x y
-2 -7
-1 -4
0 -1
1 2
2 5

Answer 2

If the function h(m) gives the number of helium balloons the machine can blow up per minute, then what is an appropriate domain for this scenario?(1 point)
Responses

All real numbers
All real numbers

All negative numbers
All negative numbers

All integers
All integers

All positive integers

Answer 3

The equation for the area of a square could be written: A=s2
(4 points)
1) Is this a linear or nonlinear function? (response area 1)

2) What are some of the values in the table where s is the side length and A is the area? (response area 2)

3) When you plot these points, is the graph increasing or decreasing? (response area 3)

4) What numbers would be included in the domain? (response area 4)

1 responses
Linear
nonlinear

2 responses
(1, 1), (2, 4), (3, 9)
(1, 2), (3, 4), (5, 6)
(1, 2), (2, 2), (3, 6)

3 responses
decreasing
increasing

4 responses
negative
positive

Answer 4

To determine whether the relationship between \( x \) and \( y \) in your table is linear or nonlinear, we can examine the changes in \( y \) corresponding to changes in \( x \).

Here are the pairs you provided:

\[
\begin{align*}
x = -2 & \Rightarrow y = -8 \\
x = -1 & \Rightarrow y = -3 \\
x = 0 & \Rightarrow y = 0 \\
x = 1 & \Rightarrow y = 1 \\
x = 2 & \Rightarrow y = 0 \\
\end{align*}
\]

Next, we can calculate the differences in \( y \) values as \( x \) increases.

\[
\begin{align*}
\Delta y \text{ from } -2 \text{ to } -1 &: -3 - (-8) = 5 \\
\Delta y \text{ from } -1 \text{ to } 0 &: 0 - (-3) = 3 \\
\Delta y \text{ from } 0 \text{ to } 1 &: 1 - 0 = 1 \\
\Delta y \text{ from } 1 \text{ to } 2 &: 0 - 1 = -1 \\
\end{align*}
\]

The differences in \( y \) (5, 3, 1, -1) are not constant, which indicates the relationship is not linear.

We can also visualize the points:

- When \( x = -2 \), \( y = -8 \)
- When \( x = -1 \), \( y = -3 \)
- When \( x = 0 \), \( y = 0 \)
- When \( x = 1 \), \( y = 1 \)
- When \( x = 2 \), \( y = 0 \)

If you were to plot these pairs on a graph, you would observe that they do not form a straight line.

Thus, the relationship represented by the table is **nonlinear**.

Answer 5

To determine whether the relationship between \( x \) and \( y \) in your table is linear or nonlinear, we can examine the changes in \( y \) as \( x \) increases.

Here are the pairs you provided:

\[
\begin{align*}
x = -2 & \Rightarrow y = -7 \\
x = -1 & \Rightarrow y = -4 \\
x = 0 & \Rightarrow y = -1 \\
x = 1 & \Rightarrow y = 2 \\
x = 2 & \Rightarrow y = 5 \\
\end{align*}
\]

Next, we can calculate the differences in \( y \) values as \( x \) increases:

\[
\begin{align*}
\Delta y \text{ from } -2 \text{ to } -1 &: -4 - (-7) = 3 \\
\Delta y \text{ from } -1 \text{ to } 0 &: -1 - (-4) = 3 \\
\Delta y \text{ from } 0 \text{ to } 1 &: 2 - (-1) = 3 \\
\Delta y \text{ from } 1 \text{ to } 2 &: 5 - 2 = 3 \\
\end{align*}
\]

The differences in \( y \) (3, 3, 3, 3) are constant, which indicates that the relationship is linear.

Additionally, we can derive the equation of the line. Since the slope (\( m \)) is constant at 3, we can express the linear relationship in the form:

\[
y = mx + b
\]

To find \( b \), we can use one of the points. For example, using the point \( (0, -1) \):

\[
-1 = 3(0) + b \\
b = -1
\]

Thus, the equation is:

\[
y = 3x - 1
\]

This confirms that the relationship is linear.

**Conclusion: The relationship represented by the table is linear.**

Answer 6

In the context of a function \( h(m) \) that represents the number of helium balloons a machine can blow up per minute, the appropriate domain would focus on the feasible and realistic values for \( m \).

Given the scenario, the number of balloons blown up per minute cannot be negative or zero, as it wouldn't make sense to have a negative number of balloons or a capacity of blowing zero balloons in a minute. Thus, the domain would consist of positive values only.

The most appropriate domain in this context would be:

**All positive integers.**

This means \( m \) can take on values such as 1, 2, 3, and so on, as these represent the practical number of balloons the machine can blow up in each minute.

Answer 7

Let's go through each of your questions step by step:

1) **Is this a linear or nonlinear function?**
The equation for the area of a square is given by \( A = s^2 \), where \( s \) is the side length. Since this equation involves squaring \( s \), it is a **nonlinear function**.

**Response: Nonlinear**

2) **What are some of the values in the table where \( s \) is the side length and \( A \) is the area?**
For various positive integer values of \( s \):

- If \( s = 1 \), then \( A = 1^2 = 1 \) → (1, 1)
- If \( s = 2 \), then \( A = 2^2 = 4 \) → (2, 4)
- If \( s = 3 \), then \( A = 3^2 = 9 \) → (3, 9)

Therefore, one valid response for the table values is:
**Response: (1, 1), (2, 4), (3, 9)**

3) **When you plot these points, is the graph increasing or decreasing?**
As \( s \) increases, the area \( A \) also increases for positive values of \( s \). Therefore, the graph is **increasing**.

**Response: Increasing**

4) **What numbers would be included in the domain?**
The side length \( s \) must be a non-negative number since a side length cannot be negative. In practical terms, the domain includes all positive numbers (since we typically don't consider a square with a side length of zero for area purposes).

**Response: Positive**

### Summarized Responses:
1. Nonlinear
2. (1, 1), (2, 4), (3, 9)
3. Increasing
4. Positive