Finding Slope From Tables, Graphs, and Points

4.7.2 Finding Slope From Tables, Graphs, and Points

4.7.2 • Finding Slope From Tables, Graphs, and Points

Activity

Recall that the slope of a line is the rate of change of the linear function. On the graph, it is the steepness of the line.

1. Find the slope of the line that connects $(0, 0)$ and $(3, 2)$

$\frac{2}{3}$

2. Find the slope of the line that connects $(4, 2)$ and $(10, 7)$

$\frac{5}{6}$

3. Find the slope of the line that connects $(1, - 2)$ and $(2, 5)$

$7$

4. Find the slope of the line that connects $(- 3, 4)$ and $(- 5, - 2)$

$3$

5. Find the slope of the line that connects $(8, 3)$ and $(10, - 9)$

$- 6$

For questions 6 – 9, use the graph to find the slope of the line that goes through the two points.

A coordinate plane with four labeled points: A at (2, 3), B at (4, negative 2), C at (negative 2, 5), and D at (negative 2, negative 3), with standard grid lines and axes shown.

6. A and B

$- 2$

7. A and D

$2$

8. B and C

$- 1$

9. C and D

$- 8$

10. Determine the slope of the equation whose values are in the given table.

$x$	1	3	5	7
$y$	5	10	15	20

This table lists four corresponding values for variables x and y.

$\frac{5}{2}$

11. Determine the slope of the equation whose values are in the given table. Simplify your answer.

$x$	2	5	8	11
$y$	4	1	-3	-6

Table displaying four corresponding numerical values for two variables, x and y.

$\frac{- 3}{3} = - 1$

Self Check

Find the slope of the line that goes through

(-1,5)

and

(6,2)

$-\frac{3}{7}$
$\frac{3}{7}$
$- \frac{7}{3}$
$- \frac{3}{5}$

Additional Resources

Slope of a Line

Here’s how to find the slope of a line from its graph using $m = \frac{r i s e}{r u n}$ .

Finding Slope from a Graph

Step 1 - Locate two points on the line whose coordinates are integers.

Step 2 - Starting with one point, sketch a right triangle, going from the first point to the second point.

Step 3 - Count the rise and the run on the legs of the triangle.

Step 4 - Take the ratio of rise to run to find the slope: $m = \frac{r i s e}{r u n}$ .

Example 1

A graph with an orange downward-sloping line starting at (0, 5) and ending at (7.5, 0) on a grid.

Step 1 - Locate two points on the line whose coordinates are integers. $(0, 5)$ and $(3, 3)$

Step 2 - Starting with one point, sketch a right triangle, going from the first point to the second point.

A graph showing a downward-sloping orange line. Rise is labeled vertically from y equals 3 to y equals 6, and run is labeled horizontally from x equals 0 to x equals 3, illustrating slope calculation.

Step 3 - Count the rise and the run on the legs of the triangle.

The rise is -2.

The run is 3.

Step 4 - Take the ratio of rise to run to find the slope: $m = \frac{r i s e}{r u n}$ .

$m = - \frac{2}{3}$

Here’s how to find the slope of a line when given two points on the line.

Finding Slope from Two Points

Step 1 - Label the points ( $x_{1}, y_{1})$ and ( $x_{2}, y_{2})$ .

Step 2 - Substitute the values into the slope formula: $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Step 3 - Simplify the slope expression.

Example 2

Use the slope formula to find the slope of the line through the points $(- 2, - 3)$ and $(- 7, 4)$ .

Step 1 - Label the points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ .

$(- 2, - 3)$ $(- 7, 4)$

$(x_{1}, y_{1})$ $(x_{2}, y_{2})$

Step 2 - Substitute into the slope formula.

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{4 - (- 3)}{- 7 - (- 2)}$

Step 3 - Simplify

$m = \frac{4 - (- 3)}{- 7 - (- 2)} = \frac{4 + 3}{- 7 + 2} = \frac{7}{- 5}$

You Try: Finding Slope from a Graph or Two Points

Find the slope of the line using the graph, then confirm your answer using the slope formula.

A straight line with negative slope crosses the y-axis at 1 and the x-axis at between 1 and 2, extending from the top left to the bottom right of the coordinate grid.

Compare your answer:

$m = - \frac{3}{5}$

Step 1 - Locate two points on the line whose coordinates are integers.

$(0, 1)$ and $(5, - 2)$

Step 2 - Starting with one point, sketch a right triangle, going from the first point to the second point.

Step 3 - Count the rise and the run on the legs of the triangle.

The rise is 3.

The run is -5.

Step 4 - Take the ratio of rise to run to find the slope: $m = \frac{r i s e}{r u n}$ .

$m = - \frac{3}{5}$

Now, using the slope formula:

Step 1 - Label the points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ .

$(0, 1)$ $(5, - 2)$

$(x_{1}, y_{1})$ $(x_{2}, y_{2})$

Step 2 - Substitute into the slope formula.

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{- 2 - 1}{5 - 0}$

Step 3 - Simplify.

$m = \frac{- 2 - 1}{5 - 0} = \frac{- 3}{5}$

The slope of the line is $- \frac{3}{5}$ .

Here’s how to find the slope of a line, given a table of values.

Example 3

Find the slope of the line whose table of values is shown:

$x$	3	5	7	9	11
$y$	4	7	10	13	16

Corresponding values for variables x and y, demonstrating a linear relationship.

Step 1 - Locate two points on the line whose coordinates are integers.

$(3, 4)$ $(5, 7)$

Step 2 - Label the points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ .

$(3, 4)$ $(5, 7)$

$(x_{1}, y_{1})$ $(x_{2}, y_{2})$

Step 3 - Substitute into the slope formula.

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{7 - 4}{5 - 3}$

Step 4 - Simplify.

$m = \frac{7 - 4}{5 - 3} = \frac{3}{2}$

Try it

Finding Slope from a Table

Find the slope of the line whose table of values is shown:

$x$	-3	-1	1	3
$y$	2	4	6	8

Corresponding x and y values, likely representing points on a linear function.

Compare your answer:

Here’s how to find the slope from a table of values:

Step 1 - Locate two points on the line whose coordinates are integers.

$(- 3, 2)$ $(- 1, 4)$

Step 2 - Label the points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ .

$(- 3, 2)$ $(- 1, 4)$

$(x_{1}, y_{1})$ $(x_{2}, y_{2})$

Step 2 - Substitute into the slope formula.

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{4 - 2}{- 1 - (- 3)}$

Step 3 - Simplify.

$m = \frac{4 - 2}{- 1 - (- 3)} = \frac{2}{2} = 1$

Compare your answer:

Here’s how to find the slope from a table of values:

Step 1 - Locate two points on the line whose coordinates are integers. - Locate two points on the line whose coordinates are integers.

$(- 3, 2)$

$(- 1, 4)$

Step 2 - Label the points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ . - Label the points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ .

$(- 3, 2)$

$(- 1, 4)$

$(x_{1}, y_{1})$

$(x_{2}, y_{2})$

Step 2 - Substitute into the slope formula. - Substitute into the slope formula.

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{4 - 2}{- 1 - (- 3)}$

Step 3 - Simplify. - Simplify.

$m = \frac{4 - 2}{- 1 - (- 3)} = \frac{2}{2} = 1$