1.01 - Functions Review, Inverse, and Parameters

University:
High School
Course:
High School Physics
Academic year:

2024
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31

Pages:

18
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1 1

Unit 1: Functions 1.01 Functions Review, Inverses, and Parameters •Relation where only one value of y is associated with each value of x. Function: What is it? And the function is…? • Domain & Range • Variation (inc./dec./constant) • Extrema (min/max) • Sign (positive/negative) • Intercepts (zero(s), initial value) Properties D: R: Variation: Extrema: Sign: Zero(s): Y-intercept: Properties •Switch values in each ordered pair in a relation. •Switch x and y in equation, re-arrange for y. Inverse • Ex: f(𝑥) = 2𝑥 + 1 • 𝑦 = 2𝑥 + 1 (f(x) means y) • 𝑥 = 2𝑦 + 1 • 2𝑦 = 𝑥 – 1 • 𝑦 = 12𝑥 − 12 • 𝑓 −1 𝑥 = 12𝑥 − 12 (now y is inverse) -1 Inverse f (x) • f 𝑥 = −3𝑥 + 2 You try! 2 • Ex: f(𝑥) = 𝑥 2 •𝑦 = 𝑥 (f(x) means y) 2 •𝑥 = 𝑦 • 𝑦 = ± 𝑥 (fails vert. line test!) • 𝑓 −1 𝑥 = 𝑥 (restrict domain) -1 Inverse f (x) •Graphically, 𝑓 𝑥 and −1 𝑓 𝑥 are symmetric about the line 𝑦 = 𝑥. Function – Inverse symmetry 2 • 𝑔 𝑥 = 𝑎(𝑥 − ℎ) + 𝑘 2 • 𝑓(𝑥) = 𝑥 changed with three parameters • General function transformed: • 𝑔 𝑥 = 𝑎 𝑓[𝑏 𝑥 − ℎ ] + 𝑘 Parameters: a, b, h, k • 𝑎 > 1 : vertical stretch •0 < 𝑎 < 1 : vertical compression •𝑎 < 0 : reflection about xaxis Parameters: a • 𝑏 > 1 : horizontal compression • 0 < 𝑏 < 1 : horizontal stretch • b < 0 : reflection about y-axis • for some functions, a and b will be combined into single number. Parameters: b •h < 0 : horizontal translation to left. •h > 0 : horizontal translation to right. Parameters: h •k < 0 : vertical translation down. •k > 0 : vertical translation up. Parameters: k • f(𝑥) = 2𝑥 + 1 • f 𝑥 =− 𝑥+1 2 + 1 • f 𝑥 = 3 −4 𝑥 − 2 − 6 Identify: a, b, h, k •Visions 1 •P 24: #1a,b,g,i, #4d,h, #7a Homework