Linear Functions: The Features of Parallel and Perpendicular

Parallel and perpendicular lines are everywhere in Algebra 1. In this guide, we’ll explain how to recognize them, write their equations, and solve problems step by step—with examples and free worksheets.

What Are Parallel Linear Functions?

Parallel lines are lines that never meet. They stay the same distance apart forever. In math, this means parallel linear functions have the exact same slope, but different y-intercepts. (If the y-intercepts were also the same, we would have just one line!)

If two lines are parallel, their equations will look like this:

\[ y = m_1x + b_1 \quad \text{and} \quad y = m_2x + b_2 \]

And the key rule is:

\[ m_1 = m_2,\quad b_1 \neq b_2 \]

Example:

What Are Perpendicular Linear Functions?

Perpendicular lines are lines that meet at a right angle (90°). In math, this means their slopes are negative reciprocals of each other. That is, when we multiply the two slopes together, the result is -1.

If two lines are perpendicular, their equations will look like:

\[ y = m_1x + b_1 \quad \text{and} \quad y = m_2x + b_2 \]

And the key rule is:

\[ m_1 \cdot m_2 = -1 \]

Example:

Example Problems (with Solutions): Parallel & Perpendicular Linear Functions

Example 1 – Parallel Line through a Point

Problem:

Find the equation of the line parallel to 𝑦 = 3𝑥 – 2 that goes through the point (1, 4).

Solution:

Step 1: Use the same slope (parallel means same slope)

𝑚 = 3

Step 2: Use point-slope form

𝑦 − 𝑦₁ = 𝑚(𝑥 − 𝑥₁)

𝑦 – 4 = 3(𝑥 – 1)

Step 3: Simplify

   𝑦 = 3𝑥 + 1

Final result:

𝑦 = 3𝑥 + 1

Example 2 – Perpendicular Line through a Point

Problem:

Find the equation of the line perpendicular to \(y = -\frac{1}{4}x + 6\) that passes through the point (2, 3).

Solution:

Step 1: Take the negative reciprocal of the slope:

\[m = -1 \div \left(\frac{1}{4}\right) = 4\]

Step 2: Use point-slope form:

𝑦 – 3 = 4(𝑥 – 2)

Step 3: Simplify:

𝑦 = 4𝑥 – 5

Final result:

𝑦 = 4𝑥 – 5

Summary: Keys to Parallel & Perpendicular Linear Functions

• Parallel Lines → Same slope;
• Perpendicular Lines → Slopes are negative reciprocals;
• Use point-slope form to find the equation when we are given a point.

Additional Math Topics for Algebra 1 – with Free Worksheets

• How to Represent Linear Functions: Slope-Intercept vs Point-Slope Form
• Different Forms of Linear Functions: Standard, Slope-Intercept & Point-Slope
• How to Graph and Solve a Linear Function Step by Step

Want more printable practice?

To access more printable worksheets on other math topics, visit this page.

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