March 22, Aim: To review for Quarterly #3 Homework: Study Review Materials. Do Now

Transcription

1 Aim: To review for Quarterly #3 Homework: Study Review Materials Do Now The value of Jenny's financial account has depreciated by 8% each year. If the account was worth $5000 in 2012 when she first opened it: a) Write a function, V(x), to represent the value of this account b) Find the value in the year 2018

2 Review Topic Index 1. Exponential Growth/Decay 2. Piecewise Functions (Quad/other function) 3. Graphing Transformations of Special Functions 4. Finding Roots 5. Quad. Word Problems (Max height, etc.)

3 Concept Review: (1) Exponential Growth/Decay Exponential Model Tips/Tricks 1. Don't forget to convert percents to decimals 8% (0.08) 2. Know when to add to(growth) and when to subtract from (decay) vs Watch out for years ( is 3 years, not 2017) 4. Enter entire model into graphing calculator when evaluating

4 Sample Question: (1) Exponential Growth/Decay 1) The population of Suffolk County was 45,750 in the year Since then, the population has been growing at a rate of 1.5% annually. a) Create a function, S(x), to model this change b) Find the estimated population in the year 2016 using your model

5 Sample Question: (1) Exponential Growth/Decay 2) The value of SnapChat stock has been decreasing by 12% since it's initial offering in If the original price was $25/share: a) Create a function, C(x), to model the value: b) Find the estimated value of the stock in 2020:

6 Concept Review: (2) Piecewise Functions (Quad/other) Piecewise Function: Graph consisting of two or more functions to be graphed under a given set of domains. Recommended Graphing Tips: 1) Create a mini-table for each function (This is usually easier than using normal rules) 2) Your table should only include values defined in the domain (on the right of the function) 3) Be careful of when to use arrows, open, or closed circles 4) If asked to evaluate a given x-value, look at where the graph is filled in at that point * Remember: A function that is simply a number (no x-value), is just a horizontal line

7 Sample Problem: (2) Piecewise Functions (Quad/other) March 22, 2017

8 Sample Question: (2) Piecewise Functions {-3-6 x < -1 g(x) = -x 2 + 6x x < 5 a) Find g(-1) b) Vertex of quadratic:

9 Sample Question: (2) Piecewise Functions h(x) = {5 x 2 + 4x < x 1 1 < x 7 a) Find h(2) b) Vertex of quadratic:

10 Concept Review: (3) Graphing Transformations of Special Functions Transformations Recap 1. f(x + 2) graph slides 2 units to the left 2. f(x - 4) graph slides 4 units to the right 3. f(x) + 1 graph slides 1 unit up 4. f(x) - 5 graph slides 5 units down 5. -f(x) graph reflects over x-axis 6. 2f(x) graph is vertically stretched (narrow) f(x) graph is vertically compressed (wider) Domain: permissible x-values Range: permissible y-values D: All real numbers (- < x < ) R: y 0 (0 y < )

11 Sample Question: (3) Graphing Transformations of Special Functions 1) If the parent quadratic function, f(x) = x 2, is transformed to create g(x) using the rule: g(x) = f(x + 5) - 2 a) How is the graph of f(x) transformed to create g(x)? b) Identify domain/range

12 Sample Question: (3) Graphing Transformations of Special Functions 1. Given f(x) = x, graph g(x) if: g(x) = f(x - 3) - 2 a) Describe transformation b) Identify domain/range c) A new graph, h(x), is defined by: h(x) = -3g(x). Describe in words what happens to the graph of g(x)

13 Sample Question: (3) Graphing Transformations of Special Functions 1. Given f(x) = x, graph g(x) if: g(x) = f(x + 4) + 3 a) Describe transformation b) Identify domain/range c) A new graph, h(x), is defined by: h(x) = -0.5g(x). Describe in words what happens to the graph of g(x)

14 Concept Review: (4) Finding Roots Algebraically Roots: Where the graph hits the x-axis (y = 0) Standard Form: f(x) = ax 2 + bx + c Strategies 1. Factoring/Zero Product Property: DOTS, GCF, Sum/Prod, or AC Method (A > 1) 2. Check Graphically: Observe where graph crosses x-axis (2nd + Trace, #2) or table Steps 1) Set function = 0 2) Factor (express as a product of 2 or more terms) 3) Use zero product property to set each "piece" equal to 0. 4) Solve resulting "mini" equations. 5) Check using calculator or by evaluating

15 Sample Question: (4) Finding Roots Algebrically 1. Given f(x) = x 2-7x - 60, find: a) The y-intercept of f(x) b) The roots (zeroes) of f(x)

16 Sample Question: (4) Finding Roots Algebrically 2. Let a function b be defined as b(x) = 4x 2 + 7x - 2 Algebraically determine the roots of b(x)

17 Sample Question: (4) Finding Roots Algebrically 3. Let a function c be defined as: C(x) = 5x x + 60 Algebraically determine the roots of C(x)

18 Sample Question: (4) Finding Roots Algebrically 4. Let a function, d, be defined as: d(x) = 8x 2 + 4x - 40 Algebraically determine the roots of d(x)

19 Concept Review: (5) Quadratic Word Problems Time/Height Word Problems Any reference to finding the highest elevation that an object reaches is referring to the maximum value of the parabola or more specifically: the vertex Since different letters are usually used, be careful with your axis of symmetry calculation h(t) = -16t t - 8 a b c Axis of Symmetry: t = -b 2a (This will give you the time that max. height occurs) To find the height, you need the "y" value of the vertex (h), which you can find through the use of a table or by evaluating the function at that t-value.

20 Sample Question: (5) Quadratic Word Problems Ex 1: Kemari throws a football that doesn't spiral into the air. Its height, in yards, after t seconds can be modeled by the function: h(t) = -8t t + 3. a) What is the highest point this football reaches? At what time does this occur? [Try without calculator first]

21 Sample Question: (5) Quadratic Word Problems Ex 2: Taj flips a water out of the 3rd floor of M.S. 67. The path of the water bottle can be modeled by the function: h(t) = -2.5t t a) Algebraically find the vertex of this function and interpret its meaning in this context.

22 (1) Exponential Growth/Decay 1) The value of Apple stock has been increasing by 6% since the beginning of It began that year with a price of $57 per share. a) Create a function, A(x), to model the value of the stock after x years b) Find the estimated value of the stock in 2019: