does this graph give a constant rate of change:

Question

x = 1 y = 1
x = 2 y = 4
x = 3 y = 9
x = 4 y = 16
x = 5 y = 25

Answer 1

To determine if the graph gives a constant rate of change, we need to look at the differences in y-values as x-values change.

The pairs you provided are as follows:

• For \( x = 1 \), \( y = 1 \)
• For \( x = 2 \), \( y = 4 \)
• For \( x = 3 \), \( y = 9 \)
• For \( x = 4 \), \( y = 16 \)
• For \( x = 5 \), \( y = 25 \)

Let's calculate the differences in \( y \) for each step:

1. From \( x = 1 \) to \( x = 2 \):

   • \( \Delta y = 4 - 1 = 3 \)

2. From \( x = 2 \) to \( x = 3 \):

   • \( \Delta y = 9 - 4 = 5 \)

3. From \( x = 3 \) to \( x = 4 \):

   • \( \Delta y = 16 - 9 = 7 \)

4. From \( x = 4 \) to \( x = 5 \):

   • \( \Delta y = 25 - 16 = 9 \)

Now let's summarize the changes:

• The change in \( y \) values is 3, 5, 7, and 9, which are not the same.

Since the differences are not constant, this means the graph does not give a constant rate of change. Instead, the change in \( y \) increases as \( x \) increases, indicating that the relationship is quadratic, specifically \( y = x^2 \).

Answer 2

Does the graph represent a function?

Is it linear?

Is it a increasing relationship?

Is it proportional?

Answer 3

Let's analyze each of your questions based on the given points:

1. Does the graph represent a function?

Yes, the graph represents a function because each input value (x-value) corresponds to exactly one output value (y-value). This is known as the vertical line test in which a vertical line cannot intersect the graph at more than one point. In your case, for \( x = 1, 2, 3, 4, 5 \), there is exactly one \( y \) for each \( x \).

2. Is it linear?

No, the graph is not linear. A linear function has a constant rate of change (i.e., the slope is constant). As we calculated previously, the rate of change in \( y \) is not constant (it varies: 3, 5, 7, 9), which indicates that the relationship is quadratic (it follows \( y = x^2 \)) rather than linear.

3. Is it an increasing relationship?

Yes, the relationship is increasing. As the x-values increase from 1 to 5, the corresponding y-values increase from 1 to 25. Therefore, the function is increasing on the interval provided.

4. Is it proportional?

No, the graph is not proportional. A proportional relationship means that the ratio of \( y \) to \( x \) is constant (i.e., \( y = kx \) for a constant \( k \)). In this case, the ratios of \( y \) values to their corresponding \( x \) values are not constant:

• For \( x = 1, \frac{y}{x} = \frac{1}{1} = 1 \)
• For \( x = 2, \frac{y}{x} = \frac{4}{2} = 2 \)
• For \( x = 3, \frac{y}{x} = \frac{9}{3} = 3 \)
• For \( x = 4, \frac{y}{x} = \frac{16}{4} = 4 \)
• For \( x = 5, \frac{y}{x} = \frac{25}{5} = 5 \)

Since these ratios are not constant but instead increase, the relationship is not proportional.

Summary:

• Represents a function: Yes
• Linear: No
• Increasing relationship: Yes
• Proportional: No