math hl1 - santowski 9/4/2015 math hl1 - santowski 1 lesson 11 functions & transformations
TRANSCRIPT
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Math HL1 - Santowski
04/21/23Math HL1 - Santowski1
Lesson 11Functions & Transformations
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Skills Preview
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Graph the function:
Evaluate the key points at f(-4), f(0), f(2), f(-4)
Graph
Evaluate g(-2) and g(-3)
40 if22
04 if22
1)(
2 xx
xxxf
€
g(x)=−12
f 2x+8( )−1
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Skills Preview
04/21/23Math HL1 - Santowski
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Lesson Objectives
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• Given the graph of a function y = f(x), be able to graph the transformed function
• Provide a complete analysis of the following types of graphs: quadratic, root, cubic, reciprocal, exponential
• Given the equation of y = f(x), be able to determine the new equation, new domain, and sketch the transformed function
dxcbafy
y af b x c d
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Parent Functions
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Base Function
Features Example
f(x) = x2
Vertex & axis of symmetry, pts (+2,4)
f(x) = 2x2 + 6x - 5
f(x) = √x
“vertex” , pts of (4,2) f(x) = -√(3-x)
f(x) = x3
Max & mins f(x) = x3 – 12x
f(x) = 1/x
Asymptotes and pts (1,1) and (-1,-1)
f(x) = 3 – 1/(x + 2)
f(x) = |x|
“vertex” and (+1,1) f(x) = 2|x – 3|
f(x) = 2x
Asymptote, pt (0,1) f(x) = -2x – 4 + 3
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(A) Working with Parabolas
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The key features of a parabola that will be helpful in studying transformations you should already know the vertex and the axis of symmetry as well as being able to work with any key order pairs (i.e. (1,1) or (2,4))
And since the vertex is a key point, you should be able to connect the vertex form and the process of completing the square to identifying transformations of y = x2.
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(A) Working with ParabolasGiven the parent function, f(x) = x2 as well as
the transformed quadratic g(x) = 2x2 + 6x – 5
(a) Explain how f(x) has been transformed to g(x)
(b) Sketch y = g(x), showing three key points on g(x)
(c) Sketch y = 1/g(x), showing key features of the reciprocal function
(d) Graph y = 1/g(x) on your TI-84 on the domain of (-4,2] & identify the range, max/mins, & intercepts
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(A) Working with Parabolas
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(B) Working with Root Functions
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The key features of a “sideways” parabola or a root function that will be helpful in studying transformations the “vertex” as well as being able to work with any key order pairs (i.e. (1,1) or (4,2))
So work with the function
PREDICT the transformations of f(x) = √x and sketch, labeling key points
3f x x
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(B) Working with Root Functions
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(C) Working with Rational Functions
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The key features of a rational function that will be helpful in studying transformations you should already know the asymptotes as well as being able to work with any key order pairs (i.e. (1,1) or (-1,-1))
So work with the function
PREDICT the transformations of f(x) = 1/x
1( ) 3
2f x
x
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(C) Working with Rational Functions
Working with g(x) = 1/x and
(a) Explain how f(x) must be transformed to get back to g(x)
(b) Sketch y = f(x), showing three key points on f(x)
(c) Sketch y = f-1(x), showing key features of the inverse function
(d) Graph y = [1/g(x)]2 on your TI-84 on the domain of (-5,5] & identify the range, max/mins, & intercepts & asymptotes
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1( ) 3
2f x
x
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(C) Working with Rational Functions
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(D) Working with Cubics f(x) = x3
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The key features of a cubic function that will be helpful in studying transformations you should know the maximums & minimum (extrema, turning points) and roots as well as being able to work with any key order pairs (i.e. (1,1) or (-1,-1))
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(D) Working with Cubics f(x) = x3
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If f(x) = x3 – 12x,
(a) Show that f(x) is an even function. Find zeroes & sketch f(x)
(b) Graph g(x) = ½ f(x – 2) + 4 (c) Determine the cubic equation for y = g(x)(d) Solve g(x) = 0
(e) CA: Graph and analyze on xER:(find asymptotes, roots, intervals of increase/decrease)
€
y=g(x)
x2 −4
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(D) Working with Cubics f(x) = x3
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So if f(x) = x3 – 12x, then graph g(x) = ½ f(x – 2) + 4
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Key Features of f(x) = 2x
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The key features of an exponential function that will be helpful in studying transformations you should know the asymptote and y-intercept as well as being able to work with any key order pairs i.e. (1,2) or (-1, ½ ) or (2,4)
So work with the function y = -2x-4 + 3
PREDICT the transformations of f(x) = 2x
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Key Features of f(x) = 2x
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Key Features of f(x) = 2x
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Culminating Assessment
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Of course, you may have the skill set from your previous math studies to demonstrate the required skills & concepts
Complete the following calculator inactive “quiz”
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Q1
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Sketch the graph of f(x) = 1/x and then sketch a graph of g(x) = -2f(3x – 6) + 4 and provide a complete functional analysis of g(x) (Domain, range, asymptotes, intercepts.)
Evaluate g(19/9)
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Q2
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Given the following graph of h(x), identify the transformations of h(x) and then graph k(x) if k(x) is defined as follows:
(a) k(x) + 2 = -2h(0.5(x + 1))(b) 4k(x) + 8 = h(4-0.5x)Show a detailed “sample
calculation” of how you transformed the point A(2,1) onto its image point
Evaluate k(-1)