The image shows a partially filled coordinate grid with points labeled in the format $(x, y)$, based on an $x \to y$ table. The points given are:

• $(0, 2)$
• $(1, 9)$
• $(2, 15)$
• $(3, 21)$
• $(4, ?)$
• $(5, ?)$
• $(6, ?)$

Analysis

The pattern in the $y$-values appears to increase by a consistent amount as $x$ increases by 1. Here’s the change between the $y$-values provided:

• From $2$ to $9$: increase of $7$
• From $9$ to $15$: increase of $6$
• From $15$ to $21$: increase of $6$

The pattern suggests a possible linear or arithmetic pattern, but with a difference between the initial increase and the subsequent ones. To find the missing values, you might continue with the assumption that each subsequent $y$-value increases by $6$ from the last known $y$-value.

Calculation for Missing Points

Assuming the increase continues at $6$ per step:

1. For $x = 4$: $21 + 6 = 27$
2. For $x = 5$: $27 + 6 = 33$
3. For $x = 6$: $33 + 6 = 39$

Thus, the complete set of points would be:

• $(0, 2), (1, 9), (2, 15), (3, 21), (4, 27), (5, 33), (6, 39)$

Observation

This appears to form a linear relationship where the $y$-values increase by approximately $6$ after the initial change of $7$. This suggests a linear function of the form $y = mx + b$.

Would you like a detailed explanation of how to derive the linear equation, or do you have any questions about this process?

Here are some related questions to expand on this topic:

1. How can we determine if this is a linear pattern or some other type of sequence?
2. How would you find the equation of the line if these points represent a linear function?
3. What would the slope and intercept be based on the values given?
4. How can we verify that our predicted points are accurate?
5. What is the significance of the consistent increase in $y$-values?

Tip: When looking for patterns, check if the increments are consistent. If they are, the relationship is likely linear, making it easier to predict further values.