Algebra 2: Unit 1 Filled Reference | High School

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High School
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High School Algebra
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2021
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Algebra 2 – Unit 1 Linear Functions (Filled Reference) This document contains completed answers and short explanations for each section of the Unit 1 packet. Use this to check your work and understand common problem types. Math Medic – Pizza Problem (Page 7) 4 toppings cost $18.99 and 6 toppings cost $22.49. Difference in cost: 22.49 − 18.99 = 3.50 Difference in toppings: 2 Cost per topping = 3.50 ÷ 2 = $1.75 per topping Cost of 5 toppings: 18.99 + 1.75 = $20.74 Upgrade from 4 to 7 toppings (3 toppings): 3 × 1.75 = $5.25 Cost of 7 toppings: 18.99 + 5.25 = $24.24 Equation: y = 1.75x + 11.99 Slope = 1.75 → cost per topping y-intercept = 11.99 → base pizza price with 0 toppings Solving Linear Equations Distribute → Combine like terms → Move variables → Move constants → Solve. Example: 6n + 5 = 2 6n = −3 n = −1/2 No solution: variables cancel but constants don’t All real numbers: both sides become identical Word Problems Define variables first. Translate words into math. Solve and check. Example: “The product of 7 and a number, increased by 3, is 39 minus twice the number.” 7x + 3 = 39 − 2x → x = 4 Absolute Value |x − 9| = 5 creates two cases: x − 9 = 5 → x = 14 x − 9 = −5 → x = 4 Always check for extraneous solutions. Relations & Functions A function has exactly one output for each input. Vertical Line Test determines if something is a function. One-to-one functions also pass the Horizontal Line Test. Slope-Intercept Form y = mx + b m = slope (rate of change) b = y-intercept (starting value) Example: 7y = 7 − 7x → y = −x + 1 Slope = −1, y-intercept = 1 Inverse Functions Steps: Replace f(x) with y Swap x and y Solve for y Rename as f■¹(x) Example: f(x) = 2x + 1 Inverse: f■¹(x) = (x − 1)/2 Sequences Recursive: an = an−1 + d Explicit (arithmetic): an = a1 + (n−1)d Example: 7, 12, 17, ... a1 = 7, d = 5 Explicit: an = 7 + 5(n−1) Scatterplots & Regression Positive correlation: both variables increase Negative correlation: one increases, one decreases Regression equation gives best-fit line. Interpret slope and intercept in context. How to Use This PDF Use this as a reference, not something to memorize. Compare your answers first, then read explanations. If your final answer matches, you’re good — even if steps differ.