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# Pre-Cal 40S Slides October 1, 2007

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Translations, stretches, reflections, even and odd functions.

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### Pre-Cal 40S Slides October 1, 2007

1. 1. Practice what you've learned ... The coordinates of a point, A, on the graph of y = ƒ(x) are (-2, -3). What are the coordinates of it's image on each of the following graphs: The image of point B after each transformation shown above is given below as point C(n). Find the original coordinates of B. C1 (2, 3) C2 (-3, 7) C3 (5, -4) C4 (-1, 6) C5 (-4, -2)
2. 2. Consider the equation below. Which transformation do you think should be applied first? second? third? fourth? Given A(-2, -3) find the coordinates of its image under the transformation given above. The image of point B after the transformation shown above is (1, 4). Find the original coordinates of B.
3. 3. Given A(-2, -3) find the coordinates of its image under the transformation given above. The image of point B after the transformation shown above is (1, 4). Find the original coordinates of B.
4. 4. Translations y = ƒ(x - a) + b DICTIONARY The role of parameter a: a > 0 the graph shifts right a units. - the x-coordinates are increased a units. Examples a < 0 the graph shifts left a units. - the x-coordinates are decreased a units. WARNING: watch the sign of a The role of parameter b: b > 0 the graph shifts up b units. - the y-coordinates are increased b units. b < 0 the graph shifts down b units. - the y-coordinates are decreased b units.
5. 5. Stretches ... (and a wee bit about reflections) Let's start with a circle ... Let's look at some graphs ... We'll head over to fooplot.com ...
6. 6. Stretches and Compressions: DICTIONARY The role of parameter a: a > 1 the graph of ƒ(x) is stretched vertically. Examples 0 < |a| < 1 the graph of ƒ(x) is compressed vertically. - the y-coordinates of ƒ are multiplied by a. The role of parameter b: b > 1 the graph of ƒ(x) is compressed horizontally. (Everything quot;speeds upquot;) 0<|b|<1 the graph of ƒ(x) is stretched horizontally. (Everything quot;slows downquot;) - the x-coordinates are multiplied by .
7. 7. Putting it all together ... y = ƒ(x) Try these examples ... REMEMBER: stretches before translations
8. 8. Reflections DICTIONARY Vertical Reflections Horizontal Reflections Given any function ƒ(x): Given any function ƒ(x): -ƒ(x) produces a reflection in the x-axis. ƒ(-x) produces a reflection in the y-axis. y-coordinates are multiplied by (-1) x-coordinates are multiplied by (-1) Inverses: the inverse of any function ƒ(x) is (read as: quot;EFF INVERSEquot;) WARNING: undoes whatever ƒ did.
9. 9. EVEN FUNCTIONS Graphically: A function is quot;evenquot; if its graph is symmetrical about the y-axis. These functions DICTIONARY are even... These are not ... Symbolically (Algebraically) a function is quot;evenquot; IFF (if and only if) ƒ(-x) = ƒ(x) Examples: Are these functions even? 1. f(x) = x² 2. g(x) = x² + 2x f(-x) = (-x)² g(-x) = (-x)² + 2(-x) f(-x) = x² g(-x) = x² - 2x since f(-x)=f(x) since g(-x) is not equal to g(x) f is an even function g is not an even function
10. 10. ODD FUNCTIONS DICTIONARY Graphically: A function is quot;oddquot; if its graph is symmetrical about the origin. These functions These are are odd ... not ... Symbolically (Algebraically) a function is quot;oddquot; IFF (if and only if) ƒ(-x) = -ƒ(x) Examples: 1. ƒ(x) = x³ - x 2. g(x) = x³- x² ƒ(-x) = (-x)³ - (-x) g(-x) = (-x)³ - (-x)² ƒ(x) = -x³ + x g(x) = -x³ - x² -ƒ(x) = -(x³ - x) -g(x) = -(x³-x²) -ƒ(x) = -x³ + x -g(x) = -x³+ x² since ƒ(-x)= -ƒ(x) since g(-x) is not equal to -g(x) ƒ is an odd function g is not an odd function