x 2 + 8x - 12 = 0 April 18, 2016 Aim: To review for Quadratic Function Exam #1 Homework: Study Review Materials

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1 im: To review for Quadratic Function Exam #1 Homework: Study Review Materials o Now - Solve using any strategy. If irrational, express in simplest radical form x 2 + 8x - 12 = 0 Review Topic Index 1. Transformations of Special Functions 2. Evaluating Special Functions 3. Rational vs. Irrational 4. Solving Quadratics/Finding Roots 5. Graphs of Special Functions 6. onverting to Vertex Form (.T.S.) 7. Graphs of Quadratics 8. Quadratic Word Problems

2 oncept Review: (1) Transformations of Special Functions The same transformation rules can be applied to quadratic, cube root, and square root functions Vertical Shifts Horizontal Shifts IHOP a(x) = x - 2 b(x) = (x + 4) 2 oefficients ax 2 where a > 1, parabola will become more narrow (vertical stretch for other functions) ax 2 where a is between 0-1: parabola will become wider Negative coefficients will reflect the graph over the x - axis (open downward) Sample Question: (1) Transformations of Special Functions 1 How does the graph of f(x) = 3x 2 compare to g(x) = x 2? f(x) opens upward and g(x) opens downward f(x) is narrower than g(x) f(x) is wider than g(x) g(x) opens upward and f(x) opens downward

3 Sample Question: (1) Transformations of Special Functions 2 How does the graph of g(x) = x compare to the parent function f(x) = x g(x) is 4 units up and 1 unit to the left of f(x) g(x) is 4 units to the right and 1 unit down from f(x) g(x) is 4 units to the left and 1 unit down from f(x) g(x) is 4 units to the left and 1 unit up from f(x) oncept Review: (2) Evaluating Special Functions To find if a given coordinate lies on a graph: Use TI-84 table or substitute in x-value oes (-2, 18) lie on the graph of the function y = x 2-4x + 7? Evaluating ube Roots f(x) = 3 x - 4 f(3) f(x) = 3 (Input a 3) (What x-value creates output of 3?)

4 Sample Question: (2) Evaluating Special Functions 3 If m(x) = 3 x + 2, for what value(s) of x does m(x) = -2? 0 only 6 only -10 and 6-10 only Sample Question: (2) Evaluating Special Functions 4 Which point is not on the graph of y = x 2-7x + 11? (-3, 40) (-1, 19) (5, 1) (7, 11)

5 oncept Review: (3) Rational vs. Irrational Irrational Numbers Non-perfect radicals π Rational Numbers Integers Repeating/Terminating ecimals If it can be written as a fraction with integer values in the numerator and denominator, it is rational Sample Question: (3) Rational vs. Irrational reate an example(s) to prove each statement below 1. The sum of a rational number and an irrational number is always irrational 2. The product of a rational and an irrational number can be rational or irrational 3. The product of two irrational values can be rational or irrational

6 oncept Review: (4) Solving Quadratics/Finding Roots Roots: Where the graph hits the x-axis (y = 0) Standard Form: f(x) = x 2 + x + Vertex Form: f(x) = a(x - h) 2 + k Strategies 1. Factoring/Zero Product Property: OTS, GF, Sum/Prod, or Method ( > 1) 2. TS: Works best when = 1, "magic number" is (b/2) 2. Use Square roots to solve 3. Quad Formula: Will always work. Must be in standard form 4. Square Root Property: Works best when given vertex form or a squared binomial Sample Question: (4) Solving Quadratics/Finding Roots 5 What are the roots of the function a(x) = x 2-8x ± 17-4 ± 17 4 ± ± 2 17

7 Sample Question: (4) Solving Quadratics/Finding Roots 6 Sample Question: (4) Solving Quadratics/Finding Roots 7 Which function has the same roots as f(x) = 2x 2-1x - 15? a(x) = (2x - 3)(x + 5) b(x) = (2x - 5)(x + 3) c(x) = (x + 5)(x - 3) d(x) = (2x + 5)(x - 3)

8 oncept Review: (5) Graphs of Special Functions To graph square root graphs: 1. onstruct table of values by first finding where radicand = 0 2. That is the endpoint (smallest x-value). omain of parent: x 0 Range: y 0 3. Next, find a few more x-values that create perfect squares If you're given a domain, do not use arrows or go beyond limits To graph cube roots graphs: omain/range: ll real numbers 1. onstruct table of values by finding where radicand = 0 2. Place that point in center of table (turning point) 3. dd 2 or 3 more points on each side by finding x-values that create perfect cubes under the radical (-8, -2, -1,0, 1, 2, 8) Sample Question: (5) Graphs of Special Functions Ex 1: Graph the function b(x) = x + 2 in the domain -2 x 7 Range:

9 Sample Question: (5) Graphs of Special Functions Ex 2: Graph the function c(x) = 3 x - 1 in the domain: -7 x 9 oncept Review: (6) onverting to Vertex Form (.T.S.) Standard Form: f(x) = x 2 + x + Vertex Form: f(x) = a(x - h) 2 + k Steps for TS: 1. Make a = 1 (ivide every term by "a") 2. Move constant (c) over 3. dd (b/2) 2 to both sides to create perfect binomial 4. Move constant/values back over to make into "f(x)" form e areful: on't forget about the value you divided out in step 1. Since this isn't an equation equal to 0, but rather a function, it does not simply go away and must be preserved.

10 Sample Question: (6) onverting to Vertex Form.T.S. Ex 1: Write the function below in vertex form by completing the square f(x) = x 2-6x + 3 Sample Question: (6) onverting to Vertex Form.T.S. Ex 2: Write the function below in vertex form by completing the square g(x) = -x 2 +12x - 20

11 oncept Review: (7) Graphs of Quadratics To graph a parabola in standard form: f(x) = x 2 + x + 1. Find axis of symmetry/vertex: x = -b/2a 2. Put this x-value in center of table. hoose 2/3 x-values above and below. 3. Fill in y-values by evaluating or using TI-84 table Vertex is always a maximum (a < 0) or minimum (a > 0) value** To graph a parabola in vertex form: f(x) = a(x - h) 2 + k ; vertex is (h, k) 1. Place vertex in center of table, choose 2/3 x-values below and above vertex 2. Fill in corresponding y-values by evaluating or using TI-84 table 3. Plot points, connect with smooth, "U-shaped" curve Sample Question: (7) Graphs of Quadratics Ex 2: Graph each function indicated below. Identify max/min and roots (x) = -x 2 + 8x (x) = -(x - 4.5) (x) 1. Max/Min: 2. Roots: (x) 1. Max/Min: 2. Roots:

12 oncept Review: (8) Quadratic Word Problems Time/Height Word Problems ny 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 OS: t = -b/2a h(t) = -16t t - 8 (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 If asked to find when the object hits the ground after being thrown, this is equivalent to finding the roots of the function, or where y = 0 (observe graph) Sample Question: (8) Quadratic Word Problems Ex 1: fter the Islanders are eliminated from the playoffs, Mr. Wieckhorst throws his calculator into the air. Mr. Rice is laughing nearby and observes that the calculators height after t seconds can be modeled by the function h(t) = -2.4t t + 38, where h is the calculators height in feet. What is the highest point his calculator reaches and at what time mark does the calculator reach this height?