📐 Concept diagram

01-05 - Linear Functions

Phase: 1 | Subject: 01-05 Prerequisites: 01-04-coordinate-geometry-2d.md (gradient, Cartesian plane) Next subject: 01-06-systems-of-linear-equations.md

Learning Objectives

By the end of this subject, you will be able to:

1. Identify linear functions and understand the form y = mx + b
2. Extract gradient and y-intercept from the slope-intercept form
3. Write the equation of a line given gradient and a point
4. Graph linear functions by hand
5. Find x and y intercepts

Core Content

What is a Function?

A function is a rule that assigns each input exactly one output. We write f(x) for "the function f evaluated at x."

A linear function produces a straight line when graphed. Its rate of change (gradient) is constant.

Slope-Intercept Form: y = mx + b

This is the most common form for linear functions:

• m = gradient (slope) — how steep the line is
• b = y-intercept — where the line crosses the y-axis (when x = 0)
• x, y = any point on the line

Example: y = 2x + 1 - m = 2 (rise 2 for every run 1) - b = 1 (crosses y-axis at (0, 1))

Example: y = -3x + 5 - m = -3 (falls 3 for every run 1) - b = 5 (crosses y-axis at (0, 5))

Example: y = x - 4 - m = 1 (rise 1 for every run 1 — a 45° line) - b = -4 (crosses y-axis at (0, -4))

Finding the Equation of a Line

Method 1: Given gradient and y-intercept If m = 3 and b = -2, equation is y = 3x - 2

Method 2: Given gradient and one point If m = 4 and line passes through (2, 5):

1. Start with y = mx + b: y = 4x + b
2. Substitute the point: 5 = 4(2) + b
3. Solve: 5 = 8 + b, so b = -3
4. Equation: y = 4x - 3

Method 3: Given two points If line passes through (1, 3) and (4, 15):

1. Find gradient: m = (15 - 3)/(4 - 1) = 12/3 = 4
2. Use Method 2 with one point: y = 4x + b
3. Substitute (1, 3): 3 = 4(1) + b, so b = -1
4. Equation: y = 4x - 1

Point-Slope Form

When you know gradient m and a point (x₁, y₁):

$y - y₁ = m(x - x₁)
$

Example: Line through (3, 2) with gradient 5

y - 2 = 5(x - 3)

This can be rearranged to slope-intercept form: y - 2 = 5x - 15 y = 5x - 13

Intercepts

y-intercept: Set x = 0 For y = 2x - 6: y-intercept is -6, or point (0, -6)

x-intercept: Set y = 0 For y = 2x - 6: 0 = 2x - 6, so x = 3, or point (3, 0)

Graphing Linear Functions

Steps: 1. Find the y-intercept (b) — plot this point 2. Use the gradient (m) to find another point - m = rise/run: from the y-intercept, go up/down m units and right/left 1 unit 3. Draw a straight line through the points

Example: Graph y = -2x + 4

1. y-intercept: b = 4, so plot (0, 4)
2. Gradient m = -2 = -2/1: from (0, 4), go DOWN 2 and RIGHT 1 to (1, 2)
3. Or go UP 2 and LEFT 1 to (-1, 6)
4. Draw line through points

Common pitfall — sign of b: When the equation is y = mx - b (e.g., y = 2x - 3), the y-intercept is -3, not 3. The minus sign is part of b. Rewrite as y = mx + (-b) if it helps: y = 2x + (-3), so b = -3.

Key Terms

• 01 05 Linear Functions
• Correct: A)
• Correct: B)
• Equation: y = 3x + 4
• Example 2: Find intercepts of 2x - 3y = 12
• Finding the Equation of a Line
• Graphing Linear Functions
• Intercepts
• Method 1: Given gradient and y-intercept
• Method 2: Given gradient and one point
• Method 3: Given two points
• Point-Slope Form

Worked Examples

Example 1: Find the equation of the line with gradient 3 passing through (1, 7)

1. Form: y = mx + b = 3x + b
2. Substitute (1, 7): 7 = 3(1) + b
3. 7 = 3 + b, so b = 4
4. Equation: y = 3x + 4

Example 2: Find intercepts of 2x - 3y = 12

Rearrange to slope-intercept form first: - 3y = 2x - 12 - y = (2/3)x - 4

y-intercept: b = -4, point (0, -4) x-intercept: Set y = 0: 0 = (2/3)x - 4, so (2/3)x = 4, x = 6, point (6, 0)

Example 3: Is the line y = 5x + 1 parallel to y = 5x - 3? What about y = -1/5 x + 2?

• Parallel to y = 5x - 3? YES — both have m = 5
• Parallel to y = -1/5 x + 2? NO — gradients are 5 and -1/5 (these are actually perpendicular!)

Quiz

Q1: What does the concept of Finding the Equation of a Line primarily refer to in this subject?

A) The definition and application of Finding the Equation of a Line B) A computational error related to Finding the Equation of a Line C) A visual representation of Finding the Equation of a Line D) A historical anecdote about Finding the Equation of a Line

Correct: A)

• If you chose A: Finding the Equation of a Line is defined as: the definition and application of finding the equation of a line. The other options describe different aspects that are not the primary focus. Correct!
• If you chose B: This is incorrect. Finding the Equation of a Line is defined as: the definition and application of finding the equation of a line. The other options describe different aspects that are not the primary focus.
• If you chose C: This is incorrect. Finding the Equation of a Line is defined as: the definition and application of finding the equation of a line. The other options describe different aspects that are not the primary focus.
• If you chose D: This is incorrect. Finding the Equation of a Line is defined as: the definition and application of finding the equation of a line. The other options describe different aspects that are not the primary focus.

Q2: What is the primary purpose of Graphing Linear Functions?

A) It is used only in advanced research contexts B) It replaces all other methods in this domain C) It is primarily a historical notation system D) It is used to graphing linear functions in mathematical analysis

Correct: D)

• If you chose A: This is incorrect. Graphing Linear Functions serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose B: This is incorrect. Graphing Linear Functions serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose C: This is incorrect. Graphing Linear Functions serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose D: Graphing Linear Functions serves the purpose described in the correct answer. The other options misrepresent its role. Correct!

Q3: Which statement about Intercepts is TRUE?

A) Intercepts is not related to this subject B) Intercepts is a fundamental concept covered in this subject C) Intercepts is mentioned only as a historical footnote D) Intercepts is an advanced topic beyond this subject's scope

Correct: B)

• If you chose A: This is incorrect. Intercepts is a fundamental concept covered in this subject. This subject covers Intercepts as part of its core content.
• If you chose B: Intercepts is a fundamental concept covered in this subject. This subject covers Intercepts as part of its core content. Correct!
• If you chose C: This is incorrect. Intercepts is a fundamental concept covered in this subject. This subject covers Intercepts as part of its core content.
• If you chose D: This is incorrect. Intercepts is a fundamental concept covered in this subject. This subject covers Intercepts as part of its core content.

Q4: Based on the worked examples in this subject, what is the correct result?

A) The inverse of the correct answer B) 3. Point (3, 0) C) An unrelated numerical value D) A different result from a common mistake

Correct: B)

• If you chose A: This is incorrect. The worked examples show that the result is 3. Point (3, 0). The other options represent common errors.
• If you chose B: The worked examples show that the result is 3. Point (3, 0). The other options represent common errors. Correct!
• If you chose C: This is incorrect. The worked examples show that the result is 3. Point (3, 0). The other options represent common errors.
• If you chose D: This is incorrect. The worked examples show that the result is 3. Point (3, 0). The other options represent common errors.

Q5: How are Intercepts and Point-Slope Form related?

A) Intercepts is a special case of Point-Slope Form B) Intercepts and Point-Slope Form are completely unrelated topics C) Intercepts is the inverse of Point-Slope Form D) Intercepts and Point-Slope Form are closely related concepts

Correct: D)

• If you chose A: This is incorrect. Both Intercepts and Point-Slope Form are covered in this subject as interconnected topics.
• If you chose B: This is incorrect. Both Intercepts and Point-Slope Form are covered in this subject as interconnected topics.
• If you chose C: This is incorrect. Both Intercepts and Point-Slope Form are covered in this subject as interconnected topics.
• If you chose D: Both Intercepts and Point-Slope Form are covered in this subject as interconnected topics. Correct!

Q6: What is a common pitfall when working with What Is A Function??

A) What Is A Function? is always computed the same way in all contexts B) What Is A Function? has no common misconceptions C) A common mistake is confusing What Is A Function? with a similar concept D) The main error with What Is A Function? is using it when it is not needed

Correct: C)

• If you chose A: This is incorrect. Students often confuse What Is A Function? with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose B: This is incorrect. Students often confuse What Is A Function? with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose C: Students often confuse What Is A Function? with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
• If you chose D: This is incorrect. Students often confuse What Is A Function? with similar-sounding or related concepts. Pay attention to the precise definitions.

Q7: When should you apply Slope-Intercept Form: Y = Mx + B?

A) Avoid Slope-Intercept Form: Y = Mx + B unless explicitly instructed B) Slope-Intercept Form: Y = Mx + B is not practically useful C) Apply Slope-Intercept Form: Y = Mx + B to solve problems in this subject's domain D) Use Slope-Intercept Form: Y = Mx + B only in pure mathematics contexts

Correct: C)

• If you chose A: This is incorrect. Slope-Intercept Form: Y = Mx + B is a practical tool used throughout this subject to solve relevant problems.
• If you chose B: This is incorrect. Slope-Intercept Form: Y = Mx + B is a practical tool used throughout this subject to solve relevant problems.
• If you chose C: Slope-Intercept Form: Y = Mx + B is a practical tool used throughout this subject to solve relevant problems. Correct!
• If you chose D: This is incorrect. Slope-Intercept Form: Y = Mx + B is a practical tool used throughout this subject to solve relevant problems.

Practice Problems

1. Identify gradient and y-intercept: y = 4x - 7
   Click for answer

m = 4, b = -7

1. Find equation with m = -2 through (3, 1)
   Click for answer

y = -2x + 7

1. Find x-intercept of y = 3x - 9
   Click for answer

Set y = 0: 3x = 9, x = 3. Point (3, 0)

1. Find y-intercept of 4x + 2y = 8
   Click for answer

2y = -4x + 8, y = -2x + 4. y-intercept = 4

1. Are y = 2x + 3 and y = 2x - 5 parallel?
   Click for answer

YES — both have m = 2

1. Find equation through (0, 4) and (2, 10)
   Click for answer

m = (10-4)/(2-0) = 3. y = 3x + 4

1. What is the gradient of a line perpendicular to y = -3/4 x + 2?
   Click for answer

Perpendicular gradient = 4/3 (negative reciprocal of -3/4)

Summary

Key takeaways:

• Linear function form: y = mx + b where m = gradient, b = y-intercept
• Gradient m = rise/run = (y₂-y₁)/(x₂-x₁)
• Point-slope form: y - y₁ = m(x - x₁)
• y-intercept: set x = 0; x-intercept: set y = 0
• Parallel lines have equal gradients
• Perpendicular lines have gradients that multiply to -1
• To graph: plot y-intercept, use gradient to find another point, draw line

Pitfalls

• Misreading the sign of b in y = mx + b. When the equation is y = 2x - 3, the y-intercept is -3, not 3. The minus sign is part of b. Rewrite as y = 2x + (-3) if it helps. This is especially error-prone when the equation comes from rearranging standard form.
• Confusing x-intercept and y-intercept methods. To find the y-intercept, set x = 0. To find the x-intercept, set y = 0. Students frequently swap these or forget to set the correct variable to zero.
• Using the point-slope form incorrectly. The form is y - y₁ = m(x - x₁), not y + y₁ or y - y₁ = m(x + x₁). The signs matter. For a point (3, 2) with gradient 5, write y - 2 = 5(x - 3), not y - 2 = 5(x + 3).
• Reading gradient directly from equations not in slope-intercept form. For an equation like 2x + 3y = 12, you must rearrange to y = (-2/3)x + 4 first. The gradient is -2/3, not 2 or 3. Never read m and b from standard form.
• Misidentifying parallel vs. perpendicular lines. Parallel lines have the SAME gradient. Perpendicular lines have gradients that are NEGATIVE RECIPROCALS (multiply to -1). Students often confuse these two conditions.

Next Steps

Next up: 01-06-systems-of-linear-equations.md