Math 3 final exam review Part 1 Unit 3: Linear Programming Unit 3.2: Quadratic Functions Unit 4: Higher-Order Polynomial Functions Unit 7: Exponential & Log Functions Math 3 final exam review Part 1

Unit 3: linear programming Let x= and y= (define your variables) Inequalities Graph cHart corner points Tell your solution (full sentence)

Unit 3.2: Quadratic Functions Standard Form: 𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐 Vertex Form: 𝑦=𝑎 (𝑥−ℎ) 2 +𝑘 (h, k) is the vertex Domain: −∞, ∞ Range: [𝑘, ∞) if a>0 (−∞, 𝑘] if a<0

Unit 3.2: Quadratic Functions 𝑦= 𝑥 2 +2𝑥−8 Standard Form: 𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐 y-intercept: (0, y) Plug in 0 for x (it’s going to be the “c” value) x-intercepts/roots: (x, 0), (x, 0) Option 2: 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎 Option 1: 𝐹𝑎𝑐𝑡𝑜𝑟, 𝑠𝑒𝑡 𝑒𝑎𝑐ℎ=0 Option 3: 𝑅𝑒𝑤𝑟𝑖𝑡𝑒 𝑖𝑛 𝑣𝑒𝑟𝑡𝑒𝑥 𝑓𝑜𝑟𝑚, 𝑠𝑜𝑙𝑣𝑒 vertex: (x, y) Option 1: 𝑥= −𝑏 2𝑎 , 𝑡ℎ𝑒𝑛 𝑝𝑙𝑢𝑔 𝑖𝑛𝑡𝑜 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑡𝑜 𝑓𝑖𝑛𝑑 𝑦 Option 2: 𝑅𝑒𝑤𝑟𝑖𝑡𝑒 𝑖𝑛 𝑣𝑒𝑟𝑡𝑒𝑥 𝑓𝑜𝑟𝑚, 𝑣𝑒𝑟𝑡𝑒𝑥 𝑖𝑠 (ℎ,𝑘)

Unit 3.2: Quadratic Functions 𝑦= 𝑥 2 +2𝑥−8 Vertex Form: 𝑦=𝑎 (𝑥−ℎ) 2 +𝑘 y-intercept: (0, y) Plug in 0 for x x-intercepts/roots: (x, 0), (x, 0) Plug in 0 for y, solve (when you take the square root, don’t for get ±‼‼‼) vertex: (x, y) Vertex is (h, k)

Unit 3.2: quadratic functions Types of Roots: 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎 Rational Roots: Radicand is a perfect square: Irrational Roots: Radicand is NOT a perfect square:

Unit 3: Quadratic Functions Convert Vertex Form to Standard Form 𝑦=2(𝑥−3 ) 2 +5 𝑦=2 𝑥−3 𝑥−3 +5 𝑦=2( 𝑥 2 −6𝑥+9)+5 𝑦=(2 𝑥 2 −12𝑥+18)+5 𝑦=2 𝑥 2 −12𝑥+23 Convert Standard Form to Vertex Form 𝑦=2 𝑥 2 −12𝑥+23 𝑦−23=2 𝑥 2 −12𝑥 𝑦−23 2 = 𝑥 2 −6𝑥 𝑦−23 2 +9= 𝑥 2 −6𝑥+9 𝑦−23 2 +9=(𝑥−3)(𝑥−3) 𝑦−23 2 +9= (𝑥−3) 2 𝑦−23 2 = (𝑥−3) 2 −9 𝑦−23= 2(𝑥−3) 2 −18 𝑦= 2(𝑥−3) 2 +5

Unit 3: Quadratic Functions Focus & Directrix 𝑦= 1 4𝑝 (𝑥−ℎ ) 2 +𝑘 P is the distance from vertex to focus, will be negative if parabola opens down

Unit 4: Higher Order Polynomial Functions End Behavior Cubic, a>0 𝑎𝑠 𝑥→∞, 𝑦 𝑖𝑛𝑐 𝑎𝑠 𝑥→−∞, 𝑦 𝑑𝑒𝑐 Linear, m>0 𝑎𝑠 𝑥→∞, 𝑦 𝑖𝑛𝑐 𝑎𝑠 𝑥→−∞, 𝑦 𝑑𝑒𝑐 Cubic, a<0 𝑎𝑠 𝑥→∞, 𝑦 𝑑𝑒𝑐 𝑎𝑠 𝑥→−∞, 𝑦 𝑖𝑛𝑐 Linear, m<0 𝑎𝑠 𝑥→∞, 𝑦 𝑑𝑒𝑐 𝑎𝑠 𝑥→−∞, 𝑦 𝑖𝑛𝑐 Quadratic, a>0 𝑎𝑠 𝑥→∞, 𝑦 𝑖𝑛𝑐 𝑎𝑠 𝑥→−∞, 𝑦 𝑖𝑛𝑐 Quartic, a>0 𝑎𝑠 𝑥→∞, 𝑦 𝑖𝑛𝑐 𝑎𝑠 𝑥→−∞, 𝑦 𝑖𝑛𝑐 Quadratic, a<0 𝑎𝑠 𝑥→∞, 𝑦 𝑑𝑒𝑐 𝑎𝑠 𝑥→−∞, 𝑦 𝑑𝑒𝑐 Quartic, a<0 𝑎𝑠 𝑥→∞, 𝑦 𝑑𝑒𝑐 𝑎𝑠 𝑥→−∞, 𝑦 𝑑𝑒𝑐

Unit 4: Higher Order Polynomial functions Solving when you can’t see all x-intercepts 1st Step: How many solutions? Cubic (𝑥 3 ): 3 Quartic (𝑥 4 ): 4 Quintic (𝑥 5 ): 5

Unit 7: exponential & logarithmic functions Growth: 𝑦=𝑎 𝑏 𝑥 b>1 Decay: 𝑦=𝑎 𝑏 𝑥 0<b<1 Exponential Functions 𝑦=𝑎 𝑏 𝑥−ℎ +𝑘 Growth/Decay Factor Initial Amount Domain: −∞, ∞ Range: 𝑘, ∞ if a>0 −∞, 𝑘 if a<0 All exponential functions have asymptotes at horizontal lines (e.g. y=0 for the graphs above, as the y values will never reach 0)

Unit 7: exponential & logarithmic functions Exponential Evaluating Rate: Percent/100 Growth: 𝑦=𝑎(1+𝑟 ) 𝑥 Decay: 𝑦=𝑎(1−𝑟 ) 𝑥 Rate: Percent/100 Compound Interest: 𝑦=𝑎(1+ 𝑟 𝑛 ) 𝑛𝑥 Number of times compounded per year Compound Continuously: 𝐴=𝑃 𝑒 𝑟𝑡 Principle (same as “a”) Amount in account at end (same as “y”)

Unit 7: exponential & logarithmic functions Inverses: Reflections over y=x Inverse Functions: Functions that “undo” one another through opposite operations. For equations: Switch x and y, solve for y For graphs: Reflect over line y=x For points: Switch x and y

Unit 7: exponential & logarithmic functions Composition of Functions

Unit 7: exponential & logarithmic functions Composition of Functions with their Inverses

Unit 7: exponential & logarithmic functions With Logarithms: 2( 5) 0.3𝑥 =30 ( 5) 0.3𝑥 =15 log 5 (5 0.3𝑥 )= log 5 (15) 0.3𝑥= log⁡(15) log⁡(5) 𝑥=5.61 Exponential Solving Without Logarithms: Isolate base with exponent Take log of both sides Evaluate log 𝟓 𝟏𝟓 Solve

Unit 7: exponential & logarithmic functions Logarithm Solving Isolate logarithm Exponentiate Solve Check for extraneous solutions