Slope - Ninth Grade Math
Introduction to Slope
Slope: A measure of the steepness and direction of a line
Symbol: Usually represented by the letter $m$
Definition: The ratio of the vertical change (rise) to the horizontal change (run) between any two points on a line
Formula: $m = \frac{\text{rise}}{\text{run}}$ or $m = \frac{\text{change in } y}{\text{change in } x}$
Symbol: Usually represented by the letter $m$
Definition: The ratio of the vertical change (rise) to the horizontal change (run) between any two points on a line
Formula: $m = \frac{\text{rise}}{\text{run}}$ or $m = \frac{\text{change in } y}{\text{change in } x}$
Types of Slope:
1. Positive Slope ($m > 0$):
• Line rises from left to right
• As $x$ increases, $y$ increases
• Example: $m = 2$, $m = \frac{3}{4}$
2. Negative Slope ($m < 0$):
• Line falls from left to right
• As $x$ increases, $y$ decreases
• Example: $m = -3$, $m = -\frac{1}{2}$
3. Zero Slope ($m = 0$):
• Horizontal line
• No vertical change (rise = 0)
• Example: $y = 5$ (any horizontal line)
4. Undefined Slope:
• Vertical line
• No horizontal change (run = 0)
• Division by zero is undefined
• Example: $x = 3$ (any vertical line)
1. Positive Slope ($m > 0$):
• Line rises from left to right
• As $x$ increases, $y$ increases
• Example: $m = 2$, $m = \frac{3}{4}$
2. Negative Slope ($m < 0$):
• Line falls from left to right
• As $x$ increases, $y$ decreases
• Example: $m = -3$, $m = -\frac{1}{2}$
3. Zero Slope ($m = 0$):
• Horizontal line
• No vertical change (rise = 0)
• Example: $y = 5$ (any horizontal line)
4. Undefined Slope:
• Vertical line
• No horizontal change (run = 0)
• Division by zero is undefined
• Example: $x = 3$ (any vertical line)
| Slope Type | Value | Line Direction | Example |
|---|---|---|---|
| Positive | $m > 0$ | Rising (↗) | $m = 2$ |
| Negative | $m < 0$ | Falling (↘) | $m = -3$ |
| Zero | $m = 0$ | Horizontal (→) | $y = 4$ |
| Undefined | No value | Vertical (↑) | $x = -2$ |
1. Find the Slope of a Graph
Rise: The vertical change between two points (up is positive, down is negative)
Run: The horizontal change between two points (right is positive, left is negative)
Rise Over Run: The method of finding slope directly from a graph
Run: The horizontal change between two points (right is positive, left is negative)
Rise Over Run: The method of finding slope directly from a graph
Slope Formula (from a graph):
$$m = \frac{\text{rise}}{\text{run}}$$
where:
• Rise = vertical change (up or down)
• Run = horizontal change (left or right)
$$m = \frac{\text{rise}}{\text{run}}$$
where:
• Rise = vertical change (up or down)
• Run = horizontal change (left or right)
Steps to Find Slope from a Graph:
Step 1: Select two points on the line with clear, integer coordinates (if possible)
Step 2: Label them as Point 1 and Point 2
Step 3: Draw a right triangle connecting the two points
Step 4: Count the rise (vertical distance):
• Moving UP = positive rise
• Moving DOWN = negative rise
Step 5: Count the run (horizontal distance):
• Moving RIGHT = positive run
• Moving LEFT = negative run
Step 6: Calculate: $m = \frac{\text{rise}}{\text{run}}$
Step 7: Simplify the fraction if possible
Step 1: Select two points on the line with clear, integer coordinates (if possible)
Step 2: Label them as Point 1 and Point 2
Step 3: Draw a right triangle connecting the two points
Step 4: Count the rise (vertical distance):
• Moving UP = positive rise
• Moving DOWN = negative rise
Step 5: Count the run (horizontal distance):
• Moving RIGHT = positive run
• Moving LEFT = negative run
Step 6: Calculate: $m = \frac{\text{rise}}{\text{run}}$
Step 7: Simplify the fraction if possible
Example 1: A line passes through points $(1, 2)$ and $(4, 8)$ on a graph. Find the slope.
Visualization:
From $(1, 2)$ to $(4, 8)$:
• Rise: From $y = 2$ to $y = 8$ → move UP 6 units → rise = +6
• Run: From $x = 1$ to $x = 4$ → move RIGHT 3 units → run = +3
Calculate slope:
$m = \frac{\text{rise}}{\text{run}} = \frac{6}{3} = 2$
Answer: $m = 2$ (positive slope, line rises)
Visualization:
From $(1, 2)$ to $(4, 8)$:
• Rise: From $y = 2$ to $y = 8$ → move UP 6 units → rise = +6
• Run: From $x = 1$ to $x = 4$ → move RIGHT 3 units → run = +3
Calculate slope:
$m = \frac{\text{rise}}{\text{run}} = \frac{6}{3} = 2$
Answer: $m = 2$ (positive slope, line rises)
Example 2: A line goes through $(0, 4)$ and $(3, 0)$. Find the slope from the graph.
From $(0, 4)$ to $(3, 0)$:
• Rise: From $y = 4$ to $y = 0$ → move DOWN 4 units → rise = -4
• Run: From $x = 0$ to $x = 3$ → move RIGHT 3 units → run = +3
$m = \frac{-4}{3} = -\frac{4}{3}$
Answer: $m = -\frac{4}{3}$ (negative slope, line falls)
From $(0, 4)$ to $(3, 0)$:
• Rise: From $y = 4$ to $y = 0$ → move DOWN 4 units → rise = -4
• Run: From $x = 0$ to $x = 3$ → move RIGHT 3 units → run = +3
$m = \frac{-4}{3} = -\frac{4}{3}$
Answer: $m = -\frac{4}{3}$ (negative slope, line falls)
Example 3: A horizontal line passes through $(2, 5)$ and $(7, 5)$.
• Rise: From $y = 5$ to $y = 5$ → no change → rise = 0
• Run: From $x = 2$ to $x = 7$ → move RIGHT 5 units → run = +5
$m = \frac{0}{5} = 0$
Answer: $m = 0$ (horizontal line)
• Rise: From $y = 5$ to $y = 5$ → no change → rise = 0
• Run: From $x = 2$ to $x = 7$ → move RIGHT 5 units → run = +5
$m = \frac{0}{5} = 0$
Answer: $m = 0$ (horizontal line)
Example 4: A vertical line passes through $(3, 1)$ and $(3, 6)$.
• Rise: From $y = 1$ to $y = 6$ → move UP 5 units → rise = +5
• Run: From $x = 3$ to $x = 3$ → no change → run = 0
$m = \frac{5}{0}$ = undefined (cannot divide by zero)
Answer: Undefined slope (vertical line)
• Rise: From $y = 1$ to $y = 6$ → move UP 5 units → rise = +5
• Run: From $x = 3$ to $x = 3$ → no change → run = 0
$m = \frac{5}{0}$ = undefined (cannot divide by zero)
Answer: Undefined slope (vertical line)
Tips for Finding Slope from a Graph:
• Choose points with integer coordinates to make counting easier
• Be consistent with direction (both from left to right or right to left)
• Always simplify your final answer
• Watch the signs! Negative slope means the line falls
• Draw a right triangle to visualize rise and run
• Choose points with integer coordinates to make counting easier
• Be consistent with direction (both from left to right or right to left)
• Always simplify your final answer
• Watch the signs! Negative slope means the line falls
• Draw a right triangle to visualize rise and run
2. Find the Slope from Two Points
Slope Formula: A formula to calculate slope when given coordinates of two points
Notation: Points are written as $(x_1, y_1)$ and $(x_2, y_2)$
Notation: Points are written as $(x_1, y_1)$ and $(x_2, y_2)$
Slope Formula (Two Points):
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
where:
• $(x_1, y_1)$ = coordinates of the first point
• $(x_2, y_2)$ = coordinates of the second point
• $m$ = slope of the line
Alternative form:
$$m = \frac{\Delta y}{\Delta x} = \frac{\text{change in } y}{\text{change in } x}$$
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
where:
• $(x_1, y_1)$ = coordinates of the first point
• $(x_2, y_2)$ = coordinates of the second point
• $m$ = slope of the line
Alternative form:
$$m = \frac{\Delta y}{\Delta x} = \frac{\text{change in } y}{\text{change in } x}$$
Important Notes:
• The order of points doesn't matter: You'll get the same slope
• Be consistent: If you subtract $y_1$ from $y_2$, you must subtract $x_1$ from $x_2$
• The denominator cannot be zero (that would make slope undefined)
• Simplify your answer to lowest terms
• Keep track of negative signs carefully
• The order of points doesn't matter: You'll get the same slope
• Be consistent: If you subtract $y_1$ from $y_2$, you must subtract $x_1$ from $x_2$
• The denominator cannot be zero (that would make slope undefined)
• Simplify your answer to lowest terms
• Keep track of negative signs carefully
Steps to Find Slope from Two Points:
Step 1: Identify and label the coordinates: $(x_1, y_1)$ and $(x_2, y_2)$
Step 2: Write the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$
Step 3: Substitute the values into the formula
Step 4: Calculate the numerator: $y_2 - y_1$
Step 5: Calculate the denominator: $x_2 - x_1$
Step 6: Divide and simplify
Step 1: Identify and label the coordinates: $(x_1, y_1)$ and $(x_2, y_2)$
Step 2: Write the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$
Step 3: Substitute the values into the formula
Step 4: Calculate the numerator: $y_2 - y_1$
Step 5: Calculate the denominator: $x_2 - x_1$
Step 6: Divide and simplify
Example 1: Find the slope of the line passing through $(2, 3)$ and $(5, 9)$.
Step 1: Label the points
$(x_1, y_1) = (2, 3)$ and $(x_2, y_2) = (5, 9)$
Step 2: Apply the formula
$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 3}{5 - 2} = \frac{6}{3} = 2$
Answer: $m = 2$
Step 1: Label the points
$(x_1, y_1) = (2, 3)$ and $(x_2, y_2) = (5, 9)$
Step 2: Apply the formula
$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 3}{5 - 2} = \frac{6}{3} = 2$
Answer: $m = 2$
Example 2: Find the slope between $(-1, 4)$ and $(3, -2)$.
Solution:
$(x_1, y_1) = (-1, 4)$ and $(x_2, y_2) = (3, -2)$
$m = \frac{-2 - 4}{3 - (-1)} = \frac{-6}{3 + 1} = \frac{-6}{4} = -\frac{3}{2}$
Answer: $m = -\frac{3}{2}$
Solution:
$(x_1, y_1) = (-1, 4)$ and $(x_2, y_2) = (3, -2)$
$m = \frac{-2 - 4}{3 - (-1)} = \frac{-6}{3 + 1} = \frac{-6}{4} = -\frac{3}{2}$
Answer: $m = -\frac{3}{2}$
Example 3: Calculate the slope through $(6, -3)$ and $(-2, -3)$.
Solution:
$(x_1, y_1) = (6, -3)$ and $(x_2, y_2) = (-2, -3)$
$m = \frac{-3 - (-3)}{-2 - 6} = \frac{-3 + 3}{-8} = \frac{0}{-8} = 0$
Answer: $m = 0$ (horizontal line)
Solution:
$(x_1, y_1) = (6, -3)$ and $(x_2, y_2) = (-2, -3)$
$m = \frac{-3 - (-3)}{-2 - 6} = \frac{-3 + 3}{-8} = \frac{0}{-8} = 0$
Answer: $m = 0$ (horizontal line)
Example 4: Find the slope of the line through $(4, 1)$ and $(4, 7)$.
Solution:
$(x_1, y_1) = (4, 1)$ and $(x_2, y_2) = (4, 7)$
$m = \frac{7 - 1}{4 - 4} = \frac{6}{0}$ = undefined
Answer: Undefined (vertical line)
Solution:
$(x_1, y_1) = (4, 1)$ and $(x_2, y_2) = (4, 7)$
$m = \frac{7 - 1}{4 - 4} = \frac{6}{0}$ = undefined
Answer: Undefined (vertical line)
Example 5: Find the slope between $(-5, -2)$ and $(3, 6)$.
Solution:
$m = \frac{6 - (-2)}{3 - (-5)} = \frac{6 + 2}{3 + 5} = \frac{8}{8} = 1$
Answer: $m = 1$
Solution:
$m = \frac{6 - (-2)}{3 - (-5)} = \frac{6 + 2}{3 + 5} = \frac{8}{8} = 1$
Answer: $m = 1$
Common Mistakes to Avoid:
• Mixing up $x$ and $y$ coordinates
• Subtracting in different orders (must be consistent)
• Forgetting to change signs when subtracting negatives
• Not simplifying the final answer
• Dividing by zero and calling it "zero" instead of "undefined"
• Mixing up $x$ and $y$ coordinates
• Subtracting in different orders (must be consistent)
• Forgetting to change signs when subtracting negatives
• Not simplifying the final answer
• Dividing by zero and calling it "zero" instead of "undefined"
3. Find the Slope from a Table
Finding Slope from a Table: Using ordered pairs from a table of values to calculate slope
Key Concept: Choose any two points from the table and use the slope formula
Key Concept: Choose any two points from the table and use the slope formula
How to Identify if a Table Shows a Linear Relationship:
• Calculate the slope between several pairs of consecutive points
• If the slope is constant (the same) → Linear relationship
• If the slope varies → Not a linear relationship
For Linear Tables:
• You can choose ANY two points
• The slope will always be the same
• The slope represents the rate of change
• Calculate the slope between several pairs of consecutive points
• If the slope is constant (the same) → Linear relationship
• If the slope varies → Not a linear relationship
For Linear Tables:
• You can choose ANY two points
• The slope will always be the same
• The slope represents the rate of change
Steps to Find Slope from a Table:
Step 1: Choose two points (rows) from the table
Step 2: Write them as ordered pairs: $(x_1, y_1)$ and $(x_2, y_2)$
Step 3: Apply the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$
Step 4: Simplify your answer
Step 5: (Optional) Check with another pair of points to verify
Step 1: Choose two points (rows) from the table
Step 2: Write them as ordered pairs: $(x_1, y_1)$ and $(x_2, y_2)$
Step 3: Apply the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$
Step 4: Simplify your answer
Step 5: (Optional) Check with another pair of points to verify
Example 1: Find the slope from this table:
Choose two points: $(1, 5)$ and $(3, 11)$
$m = \frac{11 - 5}{3 - 1} = \frac{6}{2} = 3$
Verify with different points: $(2, 8)$ and $(4, 14)$
$m = \frac{14 - 8}{4 - 2} = \frac{6}{2} = 3$ ✓
Answer: $m = 3$ (constant slope, linear relationship)
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | 5 | 8 | 11 | 14 |
$m = \frac{11 - 5}{3 - 1} = \frac{6}{2} = 3$
Verify with different points: $(2, 8)$ and $(4, 14)$
$m = \frac{14 - 8}{4 - 2} = \frac{6}{2} = 3$ ✓
Answer: $m = 3$ (constant slope, linear relationship)
Example 2: Find the slope from this table:
Choose points: $(0, 10)$ and $(4, 2)$
$m = \frac{2 - 10}{4 - 0} = \frac{-8}{4} = -2$
Verify: $(2, 6)$ and $(6, -2)$
$m = \frac{-2 - 6}{6 - 2} = \frac{-8}{4} = -2$ ✓
Answer: $m = -2$
| x | 0 | 2 | 4 | 6 |
|---|---|---|---|---|
| y | 10 | 6 | 2 | -2 |
$m = \frac{2 - 10}{4 - 0} = \frac{-8}{4} = -2$
Verify: $(2, 6)$ and $(6, -2)$
$m = \frac{-2 - 6}{6 - 2} = \frac{-8}{4} = -2$ ✓
Answer: $m = -2$
Example 3: Determine the slope:
Choose points: $(-2, -5)$ and $(2, 1)$
$m = \frac{1 - (-5)}{2 - (-2)} = \frac{1 + 5}{2 + 2} = \frac{6}{4} = \frac{3}{2}$
Answer: $m = \frac{3}{2}$
| x | -2 | 0 | 2 | 4 |
|---|---|---|---|---|
| y | -5 | -2 | 1 | 4 |
$m = \frac{1 - (-5)}{2 - (-2)} = \frac{1 + 5}{2 + 2} = \frac{6}{4} = \frac{3}{2}$
Answer: $m = \frac{3}{2}$
Example 4: Find the slope (horizontal line):
Choose points: $(1, 4)$ and $(5, 4)$
$m = \frac{4 - 4}{5 - 1} = \frac{0}{4} = 0$
Answer: $m = 0$ (all y-values are the same → horizontal line)
| x | 1 | 3 | 5 | 7 |
|---|---|---|---|---|
| y | 4 | 4 | 4 | 4 |
$m = \frac{4 - 4}{5 - 1} = \frac{0}{4} = 0$
Answer: $m = 0$ (all y-values are the same → horizontal line)
Tips for Tables:
• Choose points that are easy to work with (avoid unnecessary complexity)
• Using consecutive points often makes calculation simpler
• If all y-values are the same → slope is 0 (horizontal)
• If all x-values are the same → slope is undefined (vertical)
• Always check that the slope is consistent across different point pairs
• Choose points that are easy to work with (avoid unnecessary complexity)
• Using consecutive points often makes calculation simpler
• If all y-values are the same → slope is 0 (horizontal)
• If all x-values are the same → slope is undefined (vertical)
• Always check that the slope is consistent across different point pairs
4. Find a Missing Coordinate Using Slope
Finding Missing Coordinate: Using the slope formula algebraically to find an unknown x or y value
Given Information: One complete point, slope, and one coordinate of another point
Method: Substitute known values into slope formula and solve for the unknown
Given Information: One complete point, slope, and one coordinate of another point
Method: Substitute known values into slope formula and solve for the unknown
Using the Slope Formula to Find Missing Values:
Start with: $m = \frac{y_2 - y_1}{x_2 - x_1}$
If finding missing x-coordinate:
1. Plug in $m$, $y_1$, $y_2$, and known $x$
2. Solve for the unknown $x$
If finding missing y-coordinate:
1. Plug in $m$, $x_1$, $x_2$, and known $y$
2. Solve for the unknown $y$
Start with: $m = \frac{y_2 - y_1}{x_2 - x_1}$
If finding missing x-coordinate:
1. Plug in $m$, $y_1$, $y_2$, and known $x$
2. Solve for the unknown $x$
If finding missing y-coordinate:
1. Plug in $m$, $x_1$, $x_2$, and known $y$
2. Solve for the unknown $y$
Steps to Find a Missing Coordinate:
Step 1: Write the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$
Step 2: Label what you know and identify the unknown variable
Step 3: Substitute all known values into the formula
Step 4: Cross-multiply to eliminate the fraction
Step 5: Solve the resulting equation for the unknown
Step 6: Check your answer by verifying the slope
Step 1: Write the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$
Step 2: Label what you know and identify the unknown variable
Step 3: Substitute all known values into the formula
Step 4: Cross-multiply to eliminate the fraction
Step 5: Solve the resulting equation for the unknown
Step 6: Check your answer by verifying the slope
Finding a Missing x-coordinate
Example 1: A line has slope $m = 3$ and passes through $(x, 7)$ and $(5, 13)$. Find $x$.
Step 1: Set up
$m = 3$, $(x_1, y_1) = (x, 7)$, $(x_2, y_2) = (5, 13)$
Step 2: Substitute into formula
$3 = \frac{13 - 7}{5 - x}$
Step 3: Simplify
$3 = \frac{6}{5 - x}$
Step 4: Cross-multiply
$3(5 - x) = 6$
$15 - 3x = 6$
Step 5: Solve
$-3x = 6 - 15$
$-3x = -9$
$x = 3$
Step 6: Check
Slope from $(3, 7)$ to $(5, 13)$: $m = \frac{13-7}{5-3} = \frac{6}{2} = 3$ ✓
Answer: $x = 3$
Step 1: Set up
$m = 3$, $(x_1, y_1) = (x, 7)$, $(x_2, y_2) = (5, 13)$
Step 2: Substitute into formula
$3 = \frac{13 - 7}{5 - x}$
Step 3: Simplify
$3 = \frac{6}{5 - x}$
Step 4: Cross-multiply
$3(5 - x) = 6$
$15 - 3x = 6$
Step 5: Solve
$-3x = 6 - 15$
$-3x = -9$
$x = 3$
Step 6: Check
Slope from $(3, 7)$ to $(5, 13)$: $m = \frac{13-7}{5-3} = \frac{6}{2} = 3$ ✓
Answer: $x = 3$
Example 2: Find $x$ if the slope is $-2$ and the points are $(1, 5)$ and $(x, -3)$.
Set up:
$-2 = \frac{-3 - 5}{x - 1}$
$-2 = \frac{-8}{x - 1}$
Cross-multiply:
$-2(x - 1) = -8$
$-2x + 2 = -8$
$-2x = -10$
$x = 5$
Answer: $x = 5$
Set up:
$-2 = \frac{-3 - 5}{x - 1}$
$-2 = \frac{-8}{x - 1}$
Cross-multiply:
$-2(x - 1) = -8$
$-2x + 2 = -8$
$-2x = -10$
$x = 5$
Answer: $x = 5$
Finding a Missing y-coordinate
Example 3: A line has slope $m = 4$ and passes through $(2, y)$ and $(5, 14)$. Find $y$.
Step 1: Set up
$m = 4$, $(x_1, y_1) = (2, y)$, $(x_2, y_2) = (5, 14)$
Step 2: Substitute
$4 = \frac{14 - y}{5 - 2}$
Step 3: Simplify
$4 = \frac{14 - y}{3}$
Step 4: Cross-multiply
$4(3) = 14 - y$
$12 = 14 - y$
Step 5: Solve
$y = 14 - 12$
$y = 2$
Check:
Slope from $(2, 2)$ to $(5, 14)$: $m = \frac{14-2}{5-2} = \frac{12}{3} = 4$ ✓
Answer: $y = 2$
Step 1: Set up
$m = 4$, $(x_1, y_1) = (2, y)$, $(x_2, y_2) = (5, 14)$
Step 2: Substitute
$4 = \frac{14 - y}{5 - 2}$
Step 3: Simplify
$4 = \frac{14 - y}{3}$
Step 4: Cross-multiply
$4(3) = 14 - y$
$12 = 14 - y$
Step 5: Solve
$y = 14 - 12$
$y = 2$
Check:
Slope from $(2, 2)$ to $(5, 14)$: $m = \frac{14-2}{5-2} = \frac{12}{3} = 4$ ✓
Answer: $y = 2$
Example 4: Find $y$ if the slope is $-\frac{1}{2}$ through $(-4, 6)$ and $(0, y)$.
Set up:
$-\frac{1}{2} = \frac{y - 6}{0 - (-4)}$
$-\frac{1}{2} = \frac{y - 6}{4}$
Cross-multiply:
$-\frac{1}{2}(4) = y - 6$
$-2 = y - 6$
$y = 4$
Answer: $y = 4$
Set up:
$-\frac{1}{2} = \frac{y - 6}{0 - (-4)}$
$-\frac{1}{2} = \frac{y - 6}{4}$
Cross-multiply:
$-\frac{1}{2}(4) = y - 6$
$-2 = y - 6$
$y = 4$
Answer: $y = 4$
Example 5: The slope is $\frac{3}{4}$ through $(8, y)$ and $(4, 3)$. Find $y$.
Solution:
$\frac{3}{4} = \frac{3 - y}{4 - 8}$
$\frac{3}{4} = \frac{3 - y}{-4}$
Cross-multiply:
$\frac{3}{4}(-4) = 3 - y$
$-3 = 3 - y$
$y = 6$
Answer: $y = 6$
Solution:
$\frac{3}{4} = \frac{3 - y}{4 - 8}$
$\frac{3}{4} = \frac{3 - y}{-4}$
Cross-multiply:
$\frac{3}{4}(-4) = 3 - y$
$-3 = 3 - y$
$y = 6$
Answer: $y = 6$
Important Tips:
• Be careful with signs when substituting negative coordinates
• Remember: subtracting a negative becomes addition
• Always cross-multiply to eliminate fractions
• Check your answer by calculating the slope with both complete points
• Keep track of which point is $(x_1, y_1)$ and which is $(x_2, y_2)$
• Be careful with signs when substituting negative coordinates
• Remember: subtracting a negative becomes addition
• Always cross-multiply to eliminate fractions
• Check your answer by calculating the slope with both complete points
• Keep track of which point is $(x_1, y_1)$ and which is $(x_2, y_2)$
Quick Reference Guide
Main Slope Formula:
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}} = \frac{\text{change in } y}{\text{change in } x}$$
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}} = \frac{\text{change in } y}{\text{change in } x}$$
Types of Slope:
• Positive slope: $m > 0$ → line rises ↗
• Negative slope: $m < 0$ → line falls ↘
• Zero slope: $m = 0$ → horizontal line →
• Undefined slope: → vertical line ↑ (division by zero)
• Positive slope: $m > 0$ → line rises ↗
• Negative slope: $m < 0$ → line falls ↘
• Zero slope: $m = 0$ → horizontal line →
• Undefined slope: → vertical line ↑ (division by zero)
Method Summary:
From a Graph:
1. Pick two points
2. Count rise (vertical change)
3. Count run (horizontal change)
4. Calculate $m = \frac{\text{rise}}{\text{run}}$
From Two Points:
1. Label points $(x_1, y_1)$ and $(x_2, y_2)$
2. Use $m = \frac{y_2 - y_1}{x_2 - x_1}$
3. Simplify
From a Table:
1. Choose any two points from table
2. Apply slope formula
3. Verify with another pair if needed
Finding Missing Coordinate:
1. Set up slope formula with known values
2. Substitute and solve for unknown
3. Check answer
From a Graph:
1. Pick two points
2. Count rise (vertical change)
3. Count run (horizontal change)
4. Calculate $m = \frac{\text{rise}}{\text{run}}$
From Two Points:
1. Label points $(x_1, y_1)$ and $(x_2, y_2)$
2. Use $m = \frac{y_2 - y_1}{x_2 - x_1}$
3. Simplify
From a Table:
1. Choose any two points from table
2. Apply slope formula
3. Verify with another pair if needed
Finding Missing Coordinate:
1. Set up slope formula with known values
2. Substitute and solve for unknown
3. Check answer
| Method | What You Have | What You Do |
|---|---|---|
| From Graph | Visual line with points | Count rise and run, divide |
| From Two Points | Two ordered pairs | Use formula $m = \frac{y_2-y_1}{x_2-x_1}$ |
| From Table | Table of x and y values | Pick two rows, use formula |
| Missing Coordinate | Slope + partial point info | Set up equation and solve |
Special Cases to Remember:
• Horizontal Line: $y = k$ (constant) → slope = 0
• Vertical Line: $x = k$ (constant) → slope = undefined
• Line through origin: If $(0,0)$ is on line, $m = \frac{y}{x}$ for any point $(x,y)$
• Parallel lines: Same slope
• Perpendicular lines: Slopes are negative reciprocals $(m_1 \cdot m_2 = -1)$
• Horizontal Line: $y = k$ (constant) → slope = 0
• Vertical Line: $x = k$ (constant) → slope = undefined
• Line through origin: If $(0,0)$ is on line, $m = \frac{y}{x}$ for any point $(x,y)$
• Parallel lines: Same slope
• Perpendicular lines: Slopes are negative reciprocals $(m_1 \cdot m_2 = -1)$
Success Tips for Slope:
✓ Remember: Slope = Rise / Run
✓ Always simplify your final answer
✓ Be careful with negative signs
✓ Check if your slope type matches the line direction
✓ Positive slope → line rises from left to right
✓ Negative slope → line falls from left to right
✓ For missing coordinates: substitute, cross-multiply, solve
✓ Always verify your answer when possible
✓ Practice recognizing slope types visually
✓ Remember: Slope = Rise / Run
✓ Always simplify your final answer
✓ Be careful with negative signs
✓ Check if your slope type matches the line direction
✓ Positive slope → line rises from left to right
✓ Negative slope → line falls from left to right
✓ For missing coordinates: substitute, cross-multiply, solve
✓ Always verify your answer when possible
✓ Practice recognizing slope types visually