Solved Exercises on Writing Equations from Graphs in Grade 8

Master writing equations from graphs: slope-intercept form, point-slope form, and linear functions through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Slope-Intercept Form
Exercise 1
Write the equation of the line shown on the graph. The line passes through (0, 3) and (2, 7).
Definition:

Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept

Method for slope-intercept form:
  1. Identify the y-intercept (where line crosses y-axis)
  2. Calculate the slope using rise over run or the slope formula
  3. Substitute values into y = mx + b
  4. Simplify if necessary
Y-Intercept
b = 3
Slope
m = 2
Equation
y = 2x + 3
Step 1: Find the y-intercept

The line crosses the y-axis at point (0, 3)

Therefore, b = 3

Step 2: Calculate the slope

Using points (0, 3) and (2, 7):

m = (y₂ - y₁)/(x₂ - x₁) = (7 - 3)/(2 - 0) = 4/2 = 2

Step 3: Write the equation

Substituting m = 2 and b = 3 into y = mx + b

y = 2x + 3

Step 4: Verify the equation

Check with point (2, 7): y = 2(2) + 3 = 4 + 3 = 7 ✓

Check with point (0, 3): y = 2(0) + 3 = 0 + 3 = 3 ✓

y = 2x + 3
Final answer:

The equation of the line is y = 2x + 3

Applied rules:

• Slope Formula: m = (y₂ - y₁)/(x₂ - x₁)

• Y-Intercept: Point where x = 0

• Slope-Intercept Form: y = mx + b

2 Finding Slope from Graph
Exercise 2
Find the equation of the line that passes through points (-1, 4) and (3, -2).
Definition:

Slope: The ratio of vertical change (rise) to horizontal change (run) between any two points on a line

Slope
m = -3/2
Point-Slope
y - 4 = -3/2(x + 1)
Slope-Int
y = -3/2x + 5/2
Step 1: Calculate the slope

Using points (-1, 4) and (3, -2):

m = (y₂ - y₁)/(x₂ - x₁) = (-2 - 4)/(3 - (-1)) = -6/4 = -3/2

Step 2: Use point-slope form

Using point (-1, 4): y - 4 = -3/2(x - (-1))

y - 4 = -3/2(x + 1)

Step 3: Convert to slope-intercept form

y - 4 = -3/2(x + 1)

y - 4 = -3/2x - 3/2

y = -3/2x - 3/2 + 4

y = -3/2x + 5/2

Step 4: Verify with both points

Check (-1, 4): y = -3/2(-1) + 5/2 = 3/2 + 5/2 = 8/2 = 4 ✓

Check (3, -2): y = -3/2(3) + 5/2 = -9/2 + 5/2 = -4/2 = -2 ✓

y = -3/2x + 5/2
Final answer:

The equation of the line is y = -3/2x + 5/2

Applied rules:

• Slope Formula: m = (y₂ - y₁)/(x₂ - x₁)

• Point-Slope Form: y - y₁ = m(x - x₁)

• Slope-Intercept Form: y = mx + b

3 Horizontal Line
Exercise 3
Write the equation of a horizontal line that passes through the point (4, -3).
Definition:

Horizontal Line: A line with slope of 0, parallel to the x-axis, with the equation y = constant

Slope
m = 0
Y-Value
y = -3
Equation
y = -3
Step 1: Identify characteristics of horizontal line

Horizontal lines have the same y-value for all x-values

Slope of horizontal line = 0

Step 2: Determine the y-coordinate

Since the line passes through (4, -3), the y-coordinate is -3

Step 3: Write the equation

For a horizontal line, y = constant

Since y = -3 for all points on the line, the equation is y = -3

Step 4: Verify

Any point on this line has y-coordinate -3 regardless of x-value

Points like (0, -3), (4, -3), (-2, -3) all satisfy y = -3

y = -3
Final answer:

The equation of the horizontal line is y = -3

Applied rules:

• Horizontal Line Equation: y = k (where k is constant)

• Slope of Horizontal Line: m = 0

• Y-Coordinate: All points have same y-value

Equation Writing Rules and Methods
\(y = mx + b\)
Slope-Intercept Form
Point-Slope
\(y - y_1 = m(x - x_1)\)
When given point and slope
Slope Formula
\(m = \frac{y_2 - y_1}{x_2 - x_1}\)
For finding slope
Standard Form
\(Ax + By = C\)
Alternative form
Key definitions:

Slope: Measure of steepness of a line, calculated as rise over run

Y-Intercept: Point where line crosses the y-axis (x = 0)

X-Intercept: Point where line crosses the x-axis (y = 0)

Slope-Intercept Form: y = mx + b where m is slope and b is y-intercept

Point-Slope Form: y - y₁ = m(x - x₁) using a point and slope

Horizontal Line: y = constant, slope = 0

Vertical Line: x = constant, slope undefined

Complete equation writing methodology:
  1. Examine the graph: Identify key features like intercepts and direction
  2. Select appropriate form: Choose slope-intercept, point-slope, or other forms
  3. Find the slope: Calculate using rise over run or slope formula
  4. Find the y-intercept: Identify where line crosses y-axis
  5. Write the equation: Substitute values into the chosen form
  6. Verify: Check that known points satisfy the equation
Tip 1: For slope, count grid squares: rise over run gives you the slope as a fraction.
Tip 2: Horizontal lines have the form y = constant, vertical lines have the form x = constant.
Tip 3: If the line goes up from left to right, the slope is positive; if it goes down, the slope is negative.
Tip 4: Always verify your equation by substituting known points from the graph.
Common errors: Mixing up x and y coordinates, incorrect slope sign, confusing intercepts, arithmetic mistakes in calculations.
Real-world applications: Rate of change problems, linear growth models, cost functions, distance-time relationships.
Essential formulas:

• Slope: m = (y₂ - y₁)/(x₂ - x₁)

• Slope-Intercept: y = mx + b

• Point-Slope: y - y₁ = m(x - x₁)

• Horizontal Line: y = k

• Vertical Line: x = k

Solution: Exercises 4 to 5
4 Vertical Line
Exercise 4
Write the equation of a vertical line that passes through the point (-2, 5).
Definition:

Vertical Line: A line with undefined slope, parallel to the y-axis, with the equation x = constant

X-Value
x = -2
Slope
undefined
Equation
x = -2
Step 1: Identify characteristics of vertical line

Vertical lines have the same x-value for all y-values

Slope of vertical line is undefined (division by zero)

Step 2: Determine the x-coordinate

Since the line passes through (-2, 5), the x-coordinate is -2

Step 3: Write the equation

For a vertical line, x = constant

Since x = -2 for all points on the line, the equation is x = -2

Step 4: Verify

Any point on this line has x-coordinate -2 regardless of y-value

Points like (-2, 0), (-2, 5), (-2, -3) all satisfy x = -2

x = -2
Final answer:

The equation of the vertical line is x = -2

Applied rules:

• Vertical Line Equation: x = k (where k is constant)

• Slope of Vertical Line: Undefined

• X-Coordinate: All points have same x-value

5 From Graph Analysis
Exercise 5
A line passes through the points (1, 2) and (4, 8). Write the equation in slope-intercept form and identify the x and y intercepts.
Definition:

Intercepts: Points where a graph crosses the coordinate axes (x-intercept: y=0, y-intercept: x=0)

Slope
m = 2
Equation
y = 2x
Y-Intercept
(0, 0)
X-Intercept
(0, 0)
Step 1: Calculate the slope

Using points (1, 2) and (4, 8):

m = (8 - 2)/(4 - 1) = 6/3 = 2

Step 2: Use point-slope form

Using point (1, 2): y - 2 = 2(x - 1)

y - 2 = 2x - 2

y = 2x

Step 3: Find y-intercept

Set x = 0: y = 2(0) = 0

Y-intercept: (0, 0)

Step 4: Find x-intercept

Set y = 0: 0 = 2x, so x = 0

X-intercept: (0, 0)

Step 5: Verify with original points

Check (1, 2): y = 2(1) = 2 ✓

Check (4, 8): y = 2(4) = 8 ✓

y = 2x
Y-int: (0,0)
X-int: (0,0)
Final answer:

The equation is y = 2x. The y-intercept is (0, 0) and the x-intercept is (0, 0).

Applied rules:

• Slope Formula: m = (y₂ - y₁)/(x₂ - x₁)

• Point-Slope to Slope-Intercept: Algebraic manipulation

• Intercept Finding: Set opposite variable to zero

Detailed Summary: Writing Equations from Graphs Techniques
\(m = \frac{y_2 - y_1}{x_2 - x_1}, \quad y = mx + b\)
Slope and Line Equation Formulas
Comprehensive definitions:

Linear Function: A function whose graph is a straight line, with a constant rate of change

Slope: The ratio of the vertical change (rise) to the horizontal change (run) between any two points on a line

Y-Intercept: The y-coordinate of the point where the line crosses the y-axis (when x = 0)

X-Intercept: The x-coordinate of the point where the line crosses the x-axis (when y = 0)

Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept

Point-Slope Form: y - y₁ = m(x - x₁), using a point (x₁, y₁) and slope m

Standard Form: Ax + By = C, where A, B, and C are integers and A is typically positive

Rate of Change: The slope of a linear function represents the constant rate of change

Complete methodology:
  1. Graph Analysis: Examine the line's direction, steepness, and intercepts
  2. Slope Determination: Calculate using rise over run or the slope formula
  3. Intercept Identification: Locate where the line crosses the axes
  4. Form Selection: Choose the most convenient form based on available information
  5. Equation Formation: Substitute values into the selected form
  6. Verification: Check that known points satisfy the equation
  7. Conversion: Transform to required form if necessary
Tip 1: When reading slope from a graph, count grid squares: rise over run gives the slope as a fraction.
Tip 2: A line going up from left to right has a positive slope; going down has a negative slope.
Tip 3: Horizontal lines have slope 0 and equation y = constant; vertical lines have undefined slope and equation x = constant.
Tip 4: Always verify your equation by substituting coordinates of known points from the graph.
Common misconceptions: Thinking slope is run over rise instead of rise over run, confusing positive and negative slopes, mixing up x and y coordinates when calculating slope.
Memorization aids: "RISE over RUN" for slope, "Y equals MX plus B" for slope-intercept form, "Point-Slope uses a POINT and SLOPE".
Critical rules and formulas:

• Slope Formula: m = (y₂ - y₁)/(x₂ - x₁)

• Slope-Intercept: y = mx + b

• Point-Slope: y - y₁ = m(x - x₁)

• Standard Form: Ax + By = C

• Horizontal Line: y = k (slope = 0)

• Vertical Line: x = k (slope undefined)

• Y-Intercept: Set x = 0 and solve for y

• X-Intercept: Set y = 0 and solve for x

Visualizing Linear Equations: Different Forms and Their Graphs
Exercise 6: Different Linear Equation Forms
Same line represented in different forms:
Slope-intercept: y = 2x + 1
Point-slope: y - 3 = 2(x - 1)
Standard form: 2x - y = -1
All represent the same line with slope 2 and y-intercept 1

Analysis: The chart demonstrates how the same linear relationship can be expressed in different algebraic forms.

  • Slope-intercept form: y = mx + b (shows slope and y-intercept directly)
  • Point-slope form: y - y₁ = m(x - x₁) (uses a point and slope)
  • Standard form: Ax + By = C (coefficients are integers)
  • All forms represent the same line geometrically

Questions & Answers

Question: How do I determine if a line has a positive or negative slope just by looking at the graph?

Answer: Here's how to identify slope direction from a graph:

  • Positive Slope: The line rises from left to right (like going uphill). As x increases, y also increases.
  • Negative Slope: The line falls from left to right (like going downhill). As x increases, y decreases.
  • Zero Slope: The line is horizontal (flat). There's no vertical change as x changes.
  • Undefined Slope: The line is vertical. There's no horizontal change as y changes.

Think of reading the graph from left to right: if you go up, slope is positive; if you go down, slope is negative; if you go straight across, slope is zero.

Example: A line going from bottom-left to top-right has a positive slope, while a line going from top-left to bottom-right has a negative slope.

Question: What's the difference between slope-intercept form and point-slope form? When should I use each?

Answer: Here's when to use each form:

  • Slope-Intercept Form (y = mx + b): Use when you know the slope and y-intercept, or when you want to easily identify these values.
  • Point-Slope Form (y - y₁ = m(x - x₁)): Use when you know the slope and one point on the line (but not necessarily the y-intercept).

Example: If you know a line has slope 3 and passes through (0, 5), use slope-intercept: y = 3x + 5.

If you know a line has slope 3 and passes through (2, 7), use point-slope: y - 7 = 3(x - 2), then convert to slope-intercept if needed.

Both forms represent the same line; you can convert between them using algebraic manipulation.

Question: How do I find the x-intercept and y-intercept from a graph?

Answer: Here's how to find intercepts from a graph:

  • Y-Intercept: Look for the point where the line crosses the y-axis (the vertical axis). At this point, x = 0. The y-intercept is in the form (0, b).
  • X-Intercept: Look for the point where the line crosses the x-axis (the horizontal axis). At this point, y = 0. The x-intercept is in the form (a, 0).

Visual cues: The y-intercept is where the line touches the vertical axis, and the x-intercept is where the line touches the horizontal axis.

From the equation: To find y-intercept, substitute x = 0 and solve for y. To find x-intercept, substitute y = 0 and solve for x.

Note: Some lines may not have both intercepts (horizontal lines don't cross the x-axis unless they are the x-axis itself).