MathematicsChapter 6: Functions and Graphs

Linear, Constant, Absolute Value

Linear: f(x)=mx+c, graph a straight line, slope m, y-intercept c.

Learning objectives

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.

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 at the origin, range y>=0.

Mathematical form

y = 2x - 1

x = y + 12

m = 2 y-intercept

y = 2x - 1 y + 1

Where / symbols

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+1)/2. 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.

Relationship

y=mx+c

Overview

Three school graphs:

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.

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.

Examples V at origin · range y>=0 · |2x-100| example

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.

Worked example

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

In simple words

A vertical line x=2 is not y=f(x). It fails the vertical-line test as a graph of a function of x.

Key points

y=mx+c
|x| is a V
m and c
piecewise by sign

Exam preparation

Exam Focus

Quick revision

Revision Zone

Review the key facts, formulas and checks before you move on.