How to Write a Point-slope Form Equation from a Graph?

Step 1: Identify a point on the line

Step 2: Find the slope of the line

Step 3: Write the equation using the point-slope form

Writing a Point-slope Form Equation from a Graph – Examples 1

Solution:

The graph shows that the line passes through point \((0, 2)\). Therefore, \((x_1, y_1) = (0, 2)\). Consider another random point on the line such as \(-2,-1)\)

Step 2: Find the slope of the line

Using the two points \((0, 2)\) and \((-2, -1)\), you can find the slope of the line as follows:

\(m = \frac{(y_2 – y_1)}{(x_2 – x_1)}\)

\(m = \frac{(2+1)}{(0+2)}\)

\(m = \frac{3}{2}\)

Therefore, the slope of the line is \(\frac{3}{2}\).

Step 3: Write the equation using the point-slope form

Now that you have the slope and a point on the line, you can write the equation of the line in the point-slope form:

\(y – y_1 = m(x – x_1)\)
\(y +1 = \frac{3}{2}(x +2)\)

This is the equation of the line in point-slope form.

Exercises for Writing a Point-slope Form Equation from a Graph

Write the equation of the line in point-slope form.

1.

2.

1. \(\color{blue}{y\:+\:2\:=−2\left(x\:−\:1\right)}\)
2. \(\color{blue}{y\:-\:3\:=\left(-\frac{1}{2}\right)\left(x+1\right)}\)

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Frequently Asked Questions

What is point-slope form?

y – y1 = m(x – x1), where (x1, y1) is any one point on the line and m is the slope.

How do I read points from a graph?

Look for clear integer coordinate points where the line passes exactly through gridline intersections. Avoid points with fractional coordinates unless you have no choice.

How do I find slope from a graph?

Pick two clear points on the line. Compute rise/run between them — vertical change divided by horizontal change. Sign matters: rises go up, falls go down.

Walk through an example.

Line through (2, 5) and (6, 13). Slope = (13-5)/(6-2) = 8/4 = 2. Use point (2, 5): y – 5 = 2(x – 2). That is the point-slope form.

Can I use either point in point-slope form?

Yes — point-slope works with ANY point on the line. The equations look different but describe the same line.

How do I convert to slope-intercept form?

Distribute the slope and solve for y. From y – 5 = 2(x – 2): y – 5 = 2x – 4, so y = 2x + 1.

What if the y-intercept is easy to read?

Then slope-intercept form (y = mx + b) is more direct. Point-slope shines when the y-intercept is hard to identify but you have one other clear point.

What is the slope of a horizontal line?

Zero. Point-slope form gives y – y1 = 0(x – x1), which simplifies to y = y1.

What is the slope of a vertical line?

Undefined. Vertical lines cannot be written in point-slope form — they are written as x = constant.

Where do students slip when writing point-slope equations?

Sign mistakes when substituting (the formula has y – y1, so a positive y1 becomes minus when plugged in), and incorrect slope direction. Always double-check by plugging a point back in.

Related Lessons You May Like

• How to find the slope of a line
• How to write linear equations from y-intercept and slope
• How to graph linear equations
• Parallel and perpendicular lines
• How to solve systems of equations

For a workbook on slope, linear equations, and the slope-intercept form, Algebra I for Beginners walks the material from first principles. Pre-Algebra for Beginners covers the prerequisites.