college algebra acosta/karwowski. chapter 3 nonlinear functions
TRANSCRIPT
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College Algebra
Acosta/Karwowski
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CHAPTER 3 Nonlinear functions
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CHAPTER 3 SECTION 1Some basic functions and concepts
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Non linear functions
• Equation sort activity
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Analyzing functions
• Analyzing a function means to learn all you can about the function using tables, graphs, logic, and intuition
• We will look at a few simple functions and build from there
• Some basic concepts are: increasing/decreasing intervals x and y intercepts local maxima/minima actual maximum/minimum (end behavior)
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Maximum/ minimum
• Maximum – the highest point the function will ever attain
• Minimum – the lowest point the function will ever attain
• Local maxima – is the exact point where the function switches from increasing to decreasing
• Local mimima – the exact point where the function switches from decreasing to increasing
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Examples:
x
y
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Using technology to find intercepts
• When you press the trace button it automatically sets on the y – intercept
• Under 2nd trace you have a “zero” option. The x – intercepts are often referred to as the zeroes of the function – this option will locate the x-intercepts if you do it correctly – the book explains how
• Easier method is to enter y = 0 function along with your f(x). This is the x axis. You have created a system. Then use the intersect feature (#5) You do need to trace close to the intercept but you then enter 3 times and you will have the x- intercept
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Examples
• Find the intercepts for the following functions
f(x) = 3x3 + x2 – x
g(x) = | 3 – x2| - 2
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Even/odd functions
• when f(x) = f(-x) for all values of x in the domain f(x) is an even function
• An even function is symmetric across the y – axis
• When f(-x) = - f(x) for all values of x in the domain f(x) is an odd function
• An odd function has rotational symmetry around the origin
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Examples - graphically
Even odd neither
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Examples - algebraically
Even ? odd ? neither• f(x) = x2 g(x) = x3 k(x) = x + 5
• m(x) = x2 – 1 n(x) = x3 – 1 j(x) = (3+x2)3
• l(x)= (x5 – x)3
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Analyzing some basic functions
• f(x) = x• g(x) = x2
• h(x) = x3
• k(x) = |x|• r(x) = 1/x• m(x) = • n(x) =
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One – non linear relation
• x2 + y2 = 1
• Distance formula – what the equation actually says
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CHAPTER 3 - SECTION 2Transformations
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f(x) notation with variable expressions
• given f(x) = 2x + 5• What does f(3x) =• What does f(x – 7) =• What does f(x2)=
• Essentially you are creating a new function.• The new function will take on characteristics of
the old function but will also insert new characteristics from the variable expression.
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Function Families• When you create new functions based on one or more other
functions you create “families” of functions with similar characteristics
• We have 7 basic functions on which to base families• Transformations are functions formed by shifting and stretching
known functions• There are 3 types of transformations translations - shifts left, right, up, or down dilations – stretching or shrinking either vertically or horizontally rotating - turning the shape around a given pointNOTE: we will not discuss rotational transformations
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Translations
• A vertical translation occurs when you add the same amount to every y-coordinate in the function
If g(x) = f(x) + a then g(x) is a vertical translation of f(x); a units• A horizontal translation occurs when you add the
same amount to every x- coordinate in the function If g(x) = f(x – a) then g(x) is a horizontal translation of f(x); a units
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Determine the parent function and the transformation indicated- sketch both
• f(x) = (x – 1)2
• k(x) = |x| + 7
• j(x) =
• m(x) = x3 + 9
• + 4
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Dilations/flips• A vertical dilation occurs when you multiply every y-coordinate by the same number – this is
often called a scale factor - a “flip” occurs if the number is negative visually this is like sticking pins in the x-intercepts and pulling/pushing up and down on the graph If g(x) = a(f(x)) then g(x) is a vertical dilation a times “larger” than f(x) • A horizontal dilation occurs when you multiply every x – coordinate by the same number. A
“flip” occurs if the number is negative. If g(x) = f(ax) then g(x) is a horizontal dilation times the size of f(x) visually this is like sticking a pin in the y- intercept and pushing/pulling sideways
Note: It is frequently difficult to tell whether it is vertical or horizontal dilation from looking at the graph
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Determine the parent function and the transformation indicated and sketch both graphs
• k(x) = (3x)2 m(x) = 9x2
• f(x) = - x3 g(x) =
• j(x) =
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Dilations with translations
• k(x) = 4(x – 5)2
• m(x) = (2x + 5)3
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Given a graph determine its equation
•
x
y
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Given a graph determine its equation
•
x
y
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Given a graph determine its equation
•
x
y
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CH 4 - CIRCLESStandard form of equation
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Transformations/ standard form
• (x – h)2 + (y – k)2 = r2
• This textbook calls this standard form for the circle equation
• It essentially embodies a transformation on the circle where the scale factor has been factored out and put to the other side
• Thus (h,k) are the coordinates of the center of the circle and r is the radius of the circle
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Graphing circles
• (x – 5)2 + (y + 2)2 = 16
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Writing the equation
• Given center and radius simply fill in the blanks
• A circle with radius 5 and center at (-2, 5)
• Given center and a point - find radius and fill in blanks
• A circle with center at (4,8) that goes through (7, 12)
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CHAPTER 3 SECTION 3Piece wise graphing
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• Sometimes an equation restricts the values of the domain
• Sometimes circumstances restrict the values of the domain
• Ex. For sales of tickets in groups of 30 -50 tickets the price will be $9
Algebra states this problem: p(x) = 9x for 30<x<50
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Piecewise functions
• A function that is built from pieces of functions by restricting the domain of each piece so that it does not overlap any other.
• Note: sometimes the functions will connect and other times they will not.
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Examples
•
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CHAPTER 3 - SECTION 4Absolute value equations
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Absolute value equations/ inequality
• From the graph of the absolute value function we can determine the nature of all absolute value equations and inequalities
f(x) = a has two solutions c and d f(x) < a is an interval [c,d] f(x)> a is a union of 2 intervals: (-∞,c) (d,∞) (note: the absolute value graph can also be seen as a piecewise graph)
x
y
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Solving algebraically
• Isolate the absolute value• Write 2 equations • Solve both equations – write solutionEx. |2x - 3| = 2 |2x – 3|< 2 |2x – 3 |> 2
| 5 – 3x | + 5 = 12 4 - |x + 3| > - 12
| x – 2| = | 4 – 3x|