Linear function and quadratic function, Lecture notes of Mathematics

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IMAT Math

1. LINEAR AND QUADRATIC FUNCTION DAVID. PARK

Transformation of function

𝑦 = 𝑓(𝑥) can be transformed into 𝑦 = 𝑓 𝑥 − 𝑎 + 𝑏 by a translation For 𝑦 = 𝑓 𝑥 − 𝑎 + 𝑏, the graph is translated horizontally 𝑎 units and vertically 𝑏 units.

Transformation of function

Consider the graph of 𝑦 = 𝑓 𝑥 Draw the graph (a) 𝑦 = 𝑓 𝑥 + 2 (b) 𝑦 = 𝑓 𝑥 + 4

Transformation of function

For 𝑦 = 𝑝𝑓 𝑥 , 𝑝 > 0 , is a vertical stretch of 𝑦 = 𝑓 𝑥 with scale factor 𝑝 with x-axis stays the same If 𝑝 > 1 , points move further away from the x-axis. If 0 < 𝑝 < 1 , points move closer to the x-axis.

Transformation of function

Consider the graph of 𝑦 = 𝑓 𝑥 Draw the graph (a) 𝑦 = 3𝑓(𝑥) (b) 𝑦 = 𝑓(2𝑥)

Transformation of function

𝑦 = −𝑓 𝑥 reflects the graph of 𝑦 = 𝑓 𝑥 over the x-axis 𝑦 = 𝑓 −𝑥 reflects the graph of 𝑦 = 𝑓 𝑥 over the y-axis 𝑦 = −𝑓 −𝑥 reflects the graph of 𝑦 = 𝑓 𝑥 over the origin

Transformation of function

For the graph of 𝑦 = 𝑓(𝑥) given, Sketch the graph (a) 𝑦 = 2𝑓(𝑥) (b) 𝑦 = 𝑓 𝑥 + 2 (c) 𝑦 = 𝑓 2 𝑥 (d) 𝑦 = 𝑓 −𝑥

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, letters and operations. A polynomial is an algebraic expression that is made up of variables and constants using only 𝑃 𝑥 = 𝑎𝑛𝑥 𝑛

• 𝑎𝑛− 1 𝑥 𝑛− 1
• ⋯ + 𝑎 1 𝑥 + 𝑎 0

Quadratic equation

Linear function

Slope of line ∆𝑦 ∆𝑥

𝑦 2 −𝑦 1 𝑥 2 −𝑥 1 Equation of line 𝑦 = 𝑚𝑥 + 𝑏 Where 𝑚 is slope and 𝑏 is intercept The x intercept is 𝑥 = − 𝑏 𝑚

Linear function

Linear function

𝑚 1 = 𝑚 (^2) 𝑚 1 𝑚 2 = − 1

Linear function

Absolute value function

Absolute value means the distance on a number line between the graph of the number and the origin