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Linear, Constant, Absolute Value
Linear: f(x)=mx+c, graph a straight line, slope m, y-intercept c. Constant: f(x)=k, horizontal line. Absolute value: |x|=x for x>=0 and |x|=-x for x<0, V-shape at the origin, range y>=0.
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Linear, Constant, Absolute Value
What is it?
Linear: f(x)=mx+c, graph a straight line, slope m, y-intercept c. Constant: f(x)=k, horizontal line. Absolute value: |x|=x for x>=0 and |x|=-x for x<0, V-shape at the origin, range y>=0.
In simple words
- Linear: f(x)=mx+c, graph a straight line, slope m, y-intercept c.
- LINEAR AND CONSTANT: y=2x-1 has slope 2 and intercept -1.
- ABSOLUTE VALUE: Graph two rays.
- Worked-method reminder: f-1(x) = x + 12
Important terms
LINEAR, CONSTANT, ABSOLUTE VALUE: Linear: f(x)=mx+c, graph a straight line, slope m, y-intercept c. Constant: f(x)=k, horizontal line. Absolute value: |x|=x for x>=0 and |x|=-x for x<0, V-shape at the origin, range y>=0.
LINEAR AND CONSTANT: y=2x-1 has slope 2 and intercept -1. Inverse of f(x)=2x-1 is x+12. Constant y=c never uses x. Vertical lines are not functions of x.
ABSOLUTE VALUE: Graph two rays. |2x-100| at x=120 is |240-100|=140. Vertex of y=|x| is (0,0). Do not write |x|=+/-x as the definition without the cases.
What you should learn
- Explain linear, constant, absolute value using the correct definitions, conditions, and notation.
- Apply y=mx+c, |x| is a V correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Examples
- slope 2
- intercept -1
- y=c horizontal
LINEAR AND CONSTANT
y=2x-1 has slope 2 and intercept -1. Inverse of f(x)=2x-1 is x+12. Constant y=c never uses x. Vertical lines are not functions of x.
ABSOLUTE VALUE
Graph two rays. |2x-100| at x=120 is |240-100|=140. Vertex of y=|x| is (0,0). Do not write |x|=+/-x as the definition without the cases.
Recommended for Class 10
Full school-level explanation
Linear: f(x)=mx+c, graph a straight line, slope m, y-intercept c. Constant: f(x)=k, horizontal line. Absolute value: |x|=x for x>=0 and |x|=-x for x<0, V-shape at the origin, range y>=0.
Three school graphs:
WHAT — LINEAR, CONSTANT, ABSOLUTE VALUE
Linear: f(x)=mx+c, graph a straight line, slope m, y-intercept c. Constant: f(x)=k, horizontal line. Absolute value: |x|=x for x>=0 and |x|=-x for x<0, V-shape at the origin, range y>=0. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Linear, Constant, Absolute Value matters because functions describe how one quantity depends on another. Domain, range, composition, and inverse ideas are the language used for graphs and mathematical models.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through LINEAR AND CONSTANT, then ABSOLUTE VALUE. Worked: f(x) = 2x - 1. Find slope, intercept, and f^{-1}(x). slope m = 2 y-intercept = -1 y = 2x - 1 y + 1 = 2x x = y + 12 f-1(x) = x + 12 Finish by checking signs, restrictions, units, and whether the answer matches the question.
LINEAR AND CONSTANT
y=2x-1 has slope 2 and intercept -1. Inverse of f(x)=2x-1 is x+12. Constant y=c never uses x. Vertical lines are not functions of x.
Remember
- m and c
Examples
- slope 2
- intercept -1
- y=c horizontal
ABSOLUTE VALUE
Graph two rays. |2x-100| at x=120 is |240-100|=140. Vertex of y=|x| is (0,0). Do not write |x|=+/-x as the definition without the cases.
Remember
- piecewise by sign
Examples
- V at origin
- range y>=0
- |2x-100| example
Remember
- y=mx+c
- |x| is a V
- x+12
- vertical is not a function of x
- m and c
- piecewise by sign
- y=mx+c
- |x| is a V
How the ideas connect
Worked: f(x) = 2x - 1. Find slope, intercept, and f-1(x). slope m = 2 y-intercept = -1 y = 2x - 1 y + 1 = 2x x = y + 12 f-1(x) = x + 12
Deeper understanding
Go beyond memorising Linear, Constant, Absolute Value. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: f(x) = 2x - 1. Find slope, intercept, and f-1(x). slope m = 2 y-intercept = -1 y = 2x - 1 y + 1 = 2x x = y + 12 f-1(x) = x + 12
Common mistakes
Exam focus
Core Idea
LINEAR AND CONSTANT
ABSOLUTE VALUE
Worked Method
Exam focus
Quick revision
- y=mx+c
- |x| is a V
- x+12
- vertical is not a function of x
- LINEAR, CONSTANT, ABSOLUTE VALUE: Linear: f(x)=mx+c, graph a straight line, slope m, y-intercept c. Constant: f(x)=k, horizontal line.
- LINEAR AND CONSTANT: y=2x-1 has slope 2 and intercept -1. Inverse of f(x)=2x-1 is x+12. Constant y=c never uses x. Ver
A vertical line x=2 is not y=f(x). It fails the vertical-line test as a graph of a function of x.
For long answers on Linear, Constant, Absolute Value: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- y=mx+c
- |x| is a V
- x+12
- vertical is not a function of x
Important formulas
- y=mx+c
- |x| is a V
- x+12
Common exam traps
- A vertical line x=2 is not y=f(x). It fails the vertical-line test as a graph of a function of x.
- Do not treat LINEAR AND CONSTANT and ABSOLUTE VALUE as the same step; each has a different purpose.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
A vertical line x=2 is not y=f(x). It fails the vertical-line test as a graph of a function of x.
- Mistake
Do not treat LINEAR AND CONSTANT and ABSOLUTE VALUE as the same step; each has a different purpose.
- Mistake
Do not jump straight to the answer; omitted signs, brackets, restrictions, or construction reasons can change the result.
- Mistake
A vertical line x=2 is not y=f(x). It fails the vertical-line test as a graph of a function of x.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
y=mx+c
|x| is a V
x+12
vertical is not a function of x
Linear: f(x)=mx+c, graph a straight line, slope m, y-intercept c. Constant: f(x)=k, horizontal line. Absolute value: |x|=x for x>=0 and |x|=-x for x<0, V-shape
y=mx+c
- Definition
- Linear: f(x)=mx+c, graph a straight line, slope m, y-intercept c. Constant: f(x)=k, horizontal line. Absolute value: |x|=x for x>=0 and |x|=-x for x<0, V-shape
- Formula
- y=mx+c
- Example
- slope 2
- Critical warning
- A vertical line x=2 is not y=f(x). It fails the vertical-line test as a graph of a function of x.
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Exam Practice
Define
Q1. Define Linear, Constant, Absolute Value.
Explain
Q2. Explain: LINEAR AND CONSTANT.
Example
Q3. Give one clear example of Linear, Constant, Absolute Value and say why it fits.
Mistake
Q4. State one common mistake students make about Linear, Constant, Absolute Value.
Exam note
Q5. Write a short exam note on Linear, Constant, Absolute Value: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.