This document defines and provides examples of linear functions. It begins by stating the objective is to define and describe linear functions using points and equations. It then defines that a linear function is of the form f(x)=mx+b, where m is the slope and b is the y-intercept. Several examples are provided to illustrate identifying if a function is linear and calculating its slope and y-intercept. The document also discusses rewriting linear equations between the standard and slope-intercept forms.

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Linear Functions

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Objective:
 Define and describe linear function
using its points and equations

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WHAT TO KNOW??
recalling translation of English phrases to
mathematical expressions and vice versa.
Example: the sum of the squares of x and y.
Answer: (x² +y²)

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Linear function
• is defined by f(x) = mx + b,
where:
 m is the slope and;
 b is the y-intercept.
 m and b are ℜ and m ≠ 0.

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Illustrative Example 1
Is the function f defined by f(x) = 2x + 3
a linear function? If yes, determine the slope
m and the y-intercept b.

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Solution:
• Yes, the function f defined by f(x) = 2x + 3
is a linear function since the highest
exponent (degree) of x is one and it is
written in the form f(x) = mx + b.
• The slope m is 2 while the y-intercept b is 3.

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Illustrative Example 2:
Is the function g defined by g(x) = -x a
linear function? If yes, determine its slope and
y-intercept.

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Solution:
• Yes, the function g is a linear function
because it has a degree one.
• Since g(x) = -x can be written as g(x) = -1x
+ 0, its slope is -1 and y-intercept is 0.

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Illustrative Example 3
Is the function h defined by h(x) = x2 +
5x + 4 a linear function?

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Solution:
• The function h is not a linear function
because its degree (the highest exponent of
x) is 2, not 1.

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Illustrative Example 4:
Function Degree Yes No m b
f(x)= 3x+4 1 Yes 3 4

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A linear equation is an equation in 2
variables which can be written in 2 forms:
Standard Form: Ax + By = C, where A, B and
C∈ℜ, A ≠ 0 and B ≠ 0; and
Slope-Intercept Form: f(x)= y = mx + b,
where m is the slope and b is the y-intercept, m
and b∈ℜ, and m ≠ 0.

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Illustrative Example 5
 How do we rewrite the equation which is in the
Standard form of 3x – 5y = 10 in the Slope-
intercept form y = mx + b? Determine its slope
and y-intercept.

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Solution:
3x – 5y = 10
3x – 5y + (-3x) = 10 + (-3x)
-5y = -3x + 10
-1/5(-5y) = -1/5(-3x + 10)
y = 3/5x – 2
The slope is 3/5 and the y-intercept
is -2.
a
Given
Addition Property of Equality
Simplification
Multiplication Property of Equality
Simplification

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Illustrative Example 6
• How do we rewrite the slope-intercept form
y = 12 x + 3 in the Standard form Ax + By =
C?

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Solution:
y = x + 3
2(y) = 2(12x + 3)
2y = x + 6
2y + (-x) = x + 6 + (-x)
-x + 2y = 6
(-1)(-x + 2y) = (-1)(6)
x – 2y = -6
Given
Multiplication Property of Equality
Simplification
Addition Property of Equality
Simplification
Multiplication Property of Equality
Simplification

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Activity 2: SUPPLY ME!
Activity 3: SAVE ME!

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Assignment:
• Study the Systems of Linear Equations and
Inequalities in Two Variables.