Point-slope form is a way to write the equation of a line when you know one point on the line and its slope. It is especially useful because you do not need to know the y-intercept first. This form helps connect the graph of a line to its algebraic equation.
It is a key tool for modeling constant rates of change in math, science, and real-world data.
Key Facts
- Point-slope form: y - y1 = m(x - x1)
- Slope formula: m = (y2 - y1)/(x2 - x1)
- In y - y1 = m(x - x1), the point (x1, y1) lies on the line.
- Slope-intercept form: y = mx + b
- To convert to slope-intercept form, distribute m and solve for y.
- A horizontal line has slope 0 and equation y = c.
Vocabulary
- Point-slope form
- A linear equation form, y - y1 = m(x - x1), that uses a known point and slope to describe a line.
- Slope
- The rate of change of a line, found by comparing vertical change to horizontal change.
- Rise
- The vertical change between two points on a line.
- Run
- The horizontal change between two points on a line.
- Slope-intercept form
- A linear equation form, y = mx + b, where m is the slope and b is the y-intercept.
Common Mistakes to Avoid
- Switching the signs of the point coordinates is wrong because y - y1 = m(x - x1) already subtracts the coordinates. For the point (3, -2), write y + 2 = m(x - 3).
- Putting the slope in the wrong place is wrong because m must multiply the entire quantity (x - x1). Write y - y1 = m(x - x1), not y - y1 = mx - x1.
- Forgetting to distribute the slope is wrong because every term inside the parentheses must be multiplied by m. For example, y - 4 = 2(x + 1) becomes y - 4 = 2x + 2.
- Using rise/run in the wrong order is wrong because slope is vertical change divided by horizontal change. Use m = change in y/change in x, not change in x/change in y.
Practice Questions
- 1 Write the equation in point-slope form for a line with slope 3 that passes through (2, 5). Then convert it to slope-intercept form.
- 2 A line passes through (-4, 1) and (2, 13). Find its slope, write the equation in point-slope form, and simplify to y = mx + b.
- 3 Two students write equations for the same line through (1, 6) with slope -2: y - 6 = -2(x - 1) and y = -2x + 8. Explain why both equations are correct.