Understanding Slope-Intercept Form: $y = mx + b$
Hey everyone! In class today, we dove into the wonderful world of linear equations, specifically the slope-intercept form. This form, $y = mx + b$, is super useful because it tells us two very important things about a line: its slope ($m$) and its y-intercept ($b$).
Let's break it down:
- Slope ($m$): The slope tells us how steep the line is and whether it's going uphill or downhill. It's often referred to as "rise over run". A positive slope means the line goes up as you move from left to right, while a negative slope means it goes down. You can calculate slope using two points $(x_1, y_1)$ and $(x_2, y_2)$ with the formula: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$
- Y-intercept ($b$): The y-intercept is the point where the line crosses the y-axis. It's the value of $y$ when $x = 0$. So, the coordinates of the y-intercept are always $(0, b)$.
Examples
Let's look at a few examples to solidify our understanding:
- Example 1: $y = 2x + 3$
- Slope ($m$) = 2
- Y-intercept ($b$) = 3 (or the point (0, 3))
- Example 2: $y = -x + 2$
- Slope ($m$) = -1 (remember, $-x$ is the same as $-1x$)
- Y-intercept ($b$) = 2 (or the point (0, 2))
- Example 3: $y = \frac{1}{2}x - 4$
- Slope ($m$) = $\frac{1}{2}$
- Y-intercept ($b$) = -4 (or the point (0, -4))
- Example 4: $y = -2x$
- Slope ($m$) = -2
- Y-intercept ($b$) = 0 (or the point (0, 0))
- Example 5: $y = 7$
- Slope ($m$) = 0
- Y-intercept ($b$) = 7 (or the point (0, 7))
Homework
To practice what we learned, please complete the following problems:
Page 86 #16-30 even
Remember to show your work and don't hesitate to ask questions if you get stuck. You can review the class notes (see links below) and the helpful videos I made!
Class Notes
- Period 2: [Link to PDF]
- Period 6: [Link to PDF]
Helpful Videos
[Links to YouTube Videos]
Bonus Point! Don't forget to write "Meatballs" in tomorrow's warm-up for a free point on your next test!
Keep up the great work, everyone! I know you can master this!