# 1.1 Lecture Notes

## Graphing in the $xy$ plane.


- Horizontal line across middle is called the x axis.
- Vertical line in the middle is called the y axis.
- They meet at the origin.
- Points in the plane are labeled with an ordered pair $(x, y)$.


- To find, or ‘plot’ the point $(2, 3)$, begin at origin and go right 2 units, then up 3 units. 'Walk then Climb'.
- Plot $(1, 5)$, $(6, 4)$, $(2, 1)$ and $(4, 6)$.

### Points and Directions Table

| **Point (x, y)** | **Walk (Right/Left)** | **Climb (Up/Down)** |
|-------------------|-----------------------|---------------------|
| $(2, 3)$ | Right 2 units | Up 3 units |
| $(1, 5)$ | Right 1 unit | Up 5 units |
| $(6, 4)$ | Right 6 units | Up 4 units |
| $(2, 1)$ | Right 2 units | Up 1 unit |
| $(4, 6)$ | Right 4 units | Up 6 units |


- **Distance between points.**
- Given points $(x_1, y_1)$ and $(x_2, y_2)$.
- Distance between them is

$$
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$

- **Example 1:** $(1, 4)$ and $(8, 7)$.

$$
d = \sqrt{(8 - 1)^2 + (7 - 4)^2}
$$

$$
d =
$$

- **Example 2:** $(1, -4)$ and $(-3, 2)$.

- **Example 3:** $(-2,-3)$ and $(5,-1)$.

- **Midpoint between two points.**
- Given points $(x_1, y_1)$ and $(x_2, y_2)$.
- Midpoint between them is

$$
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
$$

- **Midpoint Examples.**

- Example 1: $(2,4)$ and $(6,10)$.
- Example 2: $(2,5)$ and $(3,7)$.
- Example 3: $(3,-1)$ and $(-5,-7)$.
- Example 4: $(2,-4)$ and $(-3,-6)$.
- Example 5: $(1,5)$ and $(-4,-6)$.
- Example 6: $\left(\frac{3}{2}, -2\right)$ and $\left(-\frac{5}{4}, 9\right)$.
- Example 7: $\left(-2, -\frac{4}{3}\right)$ and $\left(-7, \frac{23}{6} \right)$.
- Example 8 : $\left(\frac{8}{5}, \frac{7}{3}\right)$ and $\left(\frac{11}{10}, -\frac{14}{9}\right)$.
- Example 9: $(7.4, -4.3)$ and $(-2.2, -0.1)$.