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Graph lines using slope-intercept form
Learn step-by-step with practice exercises built right in.
The graph of a linear equation is a straight line on the coordinate plane. Every point (x, y) on the line is a solution to the equation.
For example, the equation y = 2x + 1 is graphed as a line where every point satisfies the equation. The point (2, 5) is on the line because 5 = 2(2) + 1.
The coordinate plane (also called the Cartesian plane) has:
The plane is divided into four quadrants:
Points are written as (x, y) called ordered pairs:
Find the slope and y-intercept of the line: y = 3x + 2
This is in slope-intercept form: y = mx + b
Comparing y = 3x + 2 to y = mx + b: m = 3 (slope) b = 2 (y-intercept)
Slope: 3 (or 3/1, meaning rise 3, run 1) Y-intercept: 2 (point is (0, 2))
What is the slope and y-intercept of y = 4x - 3?
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To plot (3, 4):
To plot (-2, 5):
Method: Make a table of values, plot points, draw a line
Example: Graph y = 2x - 3
Step 1: Create a table (choose x-values, calculate y)
| x | y = 2x - 3 | (x, y) |
|---|---|---|
| -1 | 2(-1) - 3 = -5 | (-1, -5) |
| 0 | 2(0) - 3 = -3 | (0, -3) |
| 1 | 2(1) - 3 = -1 | (1, -1) |
| 2 | 2(2) - 3 = 1 | (2, 1) |
| 3 | 2(3) - 3 = 3 | (3, 3) |
Step 2: Plot these points on a coordinate plane
Step 3: Draw a straight line through the points
Step 4: Add arrows at both ends (line extends forever)
Tip: You only need 2 points to draw a line, but use 3 as a check!
The most common form for graphing is slope-intercept form:
y = mx + b
Where:
Example: y = 3x + 2
Slope measures the steepness and direction of a line.
Slope = rise/run = change in y / change in x = (y₂ - y₁)/(x₂ - x₁)
Types of Slope:
Positive Slope (m > 0):
Negative Slope (m < 0):
Zero Slope (m = 0):
Undefined Slope:
Method: Start at y-intercept, use slope to find next points
Example: Graph y = (2/3)x - 1
Step 1: Identify slope and y-intercept
Step 2: Plot y-intercept Plot point (0, -1)
Step 3: Use slope to find next point From (0, -1):
Step 4: Continue or go backwards From (0, -1):
Step 5: Draw line through points
Given two points (x₁, y₁) and (x₂, y₂):
slope = (y₂ - y₁)/(x₂ - x₁)
Example: Find slope through (1, 3) and (4, 9) m = (9 - 3)/(4 - 1) m = 6/3 m = 2
The y-intercept is where the line crosses the y-axis.
To find y-intercept from an equation: Set x = 0 and solve for y
Example: Find y-intercept of 2x + 3y = 12 Set x = 0: 2(0) + 3y = 12 3y = 12 y = 4 Y-intercept: (0, 4)
The x-intercept is where the line crosses the x-axis.
To find x-intercept from an equation: Set y = 0 and solve for x
Example: Find x-intercept of 2x + 3y = 12 Set y = 0: 2x + 3(0) = 12 2x = 12 x = 6 X-intercept: (6, 0)
Method: Find x-intercept and y-intercept, draw line through them
Example: Graph 3x + 2y = 12
Step 1: Find y-intercept (set x = 0) 3(0) + 2y = 12 y = 6 Point: (0, 6)
Step 2: Find x-intercept (set y = 0) 3x + 2(0) = 12 x = 4 Point: (4, 0)
Step 3: Plot both intercepts
Step 4: Draw line through them
Step 5: Check with a third point
Standard form: Ax + By = C Where A, B, C are integers and A is positive
Example: 2x + 3y = 12
To graph from standard form:
Converting to slope-intercept form: 2x + 3y = 12 3y = -2x + 12 y = (-2/3)x + 4
Horizontal Lines:
Vertical Lines:
Parallel Lines:
Perpendicular Lines:
Product of perpendicular slopes = -1
Plotting points incorrectly Remember: (x, y) not (y, x)!
Wrong direction for slope Positive slope: up-right Negative slope: down-right
Not extending line far enough Lines go on forever - add arrows!
Confusing slope and y-intercept In y = 3x + 2: slope is 3, intercept is 2
Calculating slope incorrectly Use (y₂ - y₁)/(x₂ - x₁), keep order consistent
Example 1: Phone Plan A phone plan costs 0.10 per text. Equation: y = 0.10x + 20
Example 2: Temperature Converting Fahrenheit to Celsius: C = (5/9)(F - 32) This is a linear relationship.
Example 3: Car Depreciation A car worth 2,000/year. Equation: y = -2000x + 30000
When given a word problem:
| Form | Equation | Use |
|---|---|---|
| Slope-Intercept | y = mx + b | Easy to graph, see slope/intercept |
| Standard | Ax + By = C | Find intercepts easily |
| Point-Slope | y - y₁ = m(x - x₁) | Know slope and one point |
| Horizontal | y = k | Slope is 0 |
| Vertical | x = k | Undefined slope |
Step 1: Recognize the slope-intercept form: y = mx + b where m is the slope and b is the y-intercept
Step 2: Identify m and b in y = 4x - 3: Comparing to y = mx + b: m = 4 (the coefficient of x) b = -3 (the constant term)
Step 3: State the answers: Slope = 4 (or 4/1, meaning rise 4, run 1) y-intercept = -3 (the point (0, -3))
Answer: Slope = 4, y-intercept = -3
Graph the equation y = -2x + 4 using the slope and y-intercept.
Step 1: Identify slope and y-intercept y = -2x + 4 Slope (m) = -2 = -2/1 (down 2, right 1) Y-intercept (b) = 4
Step 2: Plot y-intercept at (0, 4)
Step 3: Use slope to find another point From (0, 4), go down 2 and right 1 to get (1, 2)
Step 4: Draw line through (0, 4) and (1, 2)
The line slopes downward from left to right.
Find the slope and y-intercept of the line
The equation is already in slope-intercept form
Compare with :
This means:
Answer: Slope = , y-intercept =
Graph the equation y = -2x + 1
Step 1: Identify slope and y-intercept: Slope m = -2 (or -2/1) y-intercept b = 1 (point (0, 1))
Step 2: Plot the y-intercept: Start at (0, 1) on the graph
Step 3: Use the slope to find another point: Slope = -2/1 means: rise -2, run 1 From (0, 1): go down 2, right 1 → (1, -1)
Step 4: Plot the second point at (1, -1)
Step 5: Draw a line through both points
Points on the line: (0, 1), (1, -1), (2, -3), etc.
Answer: A line passing through (0, 1) with slope -2
Find the slope of the line passing through points (2, 3) and (6, 11).
Use the slope formula: m = (y₂ - y₁)/(x₂ - x₁)
Point 1: (2, 3) → x₁ = 2, y₁ = 3 Point 2: (6, 11) → x₂ = 6, y₂ = 11
m = (11 - 3)/(6 - 2) m = 8/4 m = 2
The slope is 2.
Find the slope of the line passing through and
Use the slope formula:
Identify the points: and
Substitute:
Answer: The slope is
Find the slope of the line passing through (2, 5) and (6, 13)
Step 1: Use the slope formula: m = (y₂ - y₁)/(x₂ - x₁)
Step 2: Identify the coordinates: Point 1: (x₁, y₁) = (2, 5) Point 2: (x₂, y₂) = (6, 13)
Step 3: Substitute into the formula: m = (13 - 5)/(6 - 2) m = 8/4 m = 2
Step 4: Interpret the slope: For every 1 unit right, the line goes up 2 units
Answer: The slope is 2
Convert to slope-intercept form
We need to solve for to get the form
Step 1: Subtract from both sides
Step 2: Divide everything by
Answer: (slope = , y-intercept = )
Write the equation of a line with slope 4 that passes through the point (1, 5).
Use point-slope form: y - y₁ = m(x - x₁)
Given: m = 4, point (1, 5)
y - 5 = 4(x - 1)
Distribute: y - 5 = 4x - 4
Add 5 to both sides: y = 4x + 1
Answer: y = 4x + 1
Write the equation of a line with slope -3 that passes through (2, 1)
Step 1: Use point-slope form: y - y₁ = m(x - x₁)
Step 2: Substitute m = -3 and point (2, 1): y - 1 = -3(x - 2)
Step 3: Distribute the -3: y - 1 = -3x + 6
Step 4: Solve for y (slope-intercept form): y = -3x + 6 + 1 y = -3x + 7
Step 5: Check: Does (2, 1) satisfy the equation? 1 = -3(2) + 7 1 = -6 + 7 1 = 1 ✓
Answer: y = -3x + 7
Write the equation of the line passing through (-1, 4) and (3, -2)
Step 1: Find the slope: m = (y₂ - y₁)/(x₂ - x₁) m = (-2 - 4)/(3 - (-1)) m = -6/4 m = -3/2
Step 2: Use point-slope form with either point (using (-1, 4)): y - 4 = (-3/2)(x - (-1)) y - 4 = (-3/2)(x + 1)
Step 3: Distribute: y - 4 = (-3/2)x - 3/2
Step 4: Solve for y: y = (-3/2)x - 3/2 + 4 y = (-3/2)x - 3/2 + 8/2 y = (-3/2)x + 5/2
Step 5: Check with both points: Point (-1, 4): 4 = (-3/2)(-1) + 5/2 = 3/2 + 5/2 = 8/2 = 4 ✓ Point (3, -2): -2 = (-3/2)(3) + 5/2 = -9/2 + 5/2 = -4/2 = -2 ✓
Answer: y = (-3/2)x + 5/2 or y = -1.5x + 2.5
Find the equation of the line passing through (3, 7) and (5, 13) in slope-intercept form.
Step 1: Find the slope m = (13 - 7)/(5 - 3) = 6/2 = 3
Step 2: Use point-slope form with either point (using (3, 7)): y - 7 = 3(x - 3)
Step 3: Convert to slope-intercept form y - 7 = 3x - 9 y = 3x - 2
Answer: y = 3x - 2
Check with other point (5, 13): y = 3(5) - 2 = 15 - 2 = 13 ✓