Question

graph of line y=7x+13. Include and label the points: (0, 13), (1, 20), (2,27). Gridlines should only appear at whole numbers. Here all horizontal gridlines are okay but vertical gridlines should only appear at -4, -3, -2, -1, 0, 1, 2, 3, 4


Write the function of the graphed arithmetic sequence.

Answer 1

The arithmetic sequence shown by the graph has terms 13, 20, 27 for indices 0, 1, 2. The common difference is 7, so the explicit formula (indexing from 0) is

a(n) = 7n + 13, for n = 0, 1, 2, ...

(If you prefer indexing from 1 so a1 = 13, use a(n) = 7n + 6.)

If you want to reproduce the requested plot (vertical gridlines only at -4..4, horizontal gridlines at all integers, and labeled points), here is a matplotlib example you can run:

```
import numpy as np
import matplotlib.pyplot as plt

# line and labeled points
x = np.linspace(-5, 5, 400)
y = 7*x + 13
pts_x = [0, 1, 2]
pts_y = [13, 20, 27]

fig, ax = plt.subplots(figsize=(6,6))
ax.plot(x, y, color='blue', label='y = 7x + 13')
ax.scatter(pts_x, pts_y, color='red', zorder=5)
for xi, yi in zip(pts_x, pts_y):
ax.annotate(f'({xi}, {yi})', (xi, yi), textcoords='offset points', xytext=(5,5))

# ticks and gridlines: vertical gridlines only at -4..4; horizontal at every integer
ax.set_xticks([-4, -3, -2, -1, 0, 1, 2, 3, 4])
ymin = min(pts_y) - 10
ymax = max(pts_y) + 10
ax.set_yticks(range(ymin, ymax + 1)) # horizontal gridlines at every integer

ax.grid(which='major', axis='x', linestyle='--', color='lightgray') # vertical gridlines at xticks
ax.grid(which='major', axis='y', linestyle='--', color='lightgray') # horizontal gridlines at yticks

ax.set_xlim(-5, 5)
ax.set_ylim(ymin, ymax)
ax.set_xlabel('x (index)')
ax.set_ylabel('a(x)')
ax.legend()
plt.show()
```

Answer (sequence function): a(n) = 7n + 13 (for n = 0,1,2, ...).