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1.5 Graphical Transformations
Represent translations algebraically and graphically
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Consider this… How is the graph (x – 2)2 + (y+1)2 = 16 related to the graph of x2 + y2 = 16?
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Some change is good!! Transformations - functions that map real numbers to real numbers Rigid transformations – leave the size and shape of a graph unchanged, include horizontal translations, vertical translations, reflections or any combination of these. Non-rigid transformations – generally distort the shape of a graph, include horizontal or vertical stretches and shrinks.
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Vertical and Horizontal Translations
1.5 Graphical Transformations Vertical and Horizontal Translations Vertical translation – shift of the graph up or down in the coordinate plane Horizontal translation – shift of the graph left or right in the coordinate plane
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Exploration #1 Complete the activity on p. 132
No talking – first 4 min. You will be able to discuss with classmates the last 2 min.
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Translations Let c be a positive real number. Then the following transformations result in translations of the graph of y = f(x) Horizontal translations y = f(x – c) a translation to the right by c units y = f(x + c) a translation to the left by c units Vertical translations y = f(x) + c a translation up by c units y = f(x) – c a translation down by c units
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Ex 1 Describe the graph of y = |x| can be transformed to the graph of the given function:
a) y = |x – 4| b) y = |x| + 2
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Reflections, Stretches, and Shrinks
Represent reflections, stretches, and shrinks of functions algebraically and graphically
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Graph in the Mirror!! Reflections – the graphs of two functions are symmetric with respect to some line Complete Exploration #2 on p. 134 First 6 min (No Talking) Last 2 min (Discuss with a neighbor)
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Reflections Over the x-axis – flips the graph of a function over the x-axis Symbolically (x,y) (x,-y) Over the y-axis – flips the graph of a function over the y-axis Symbolically (x,y) (-x,y) Over the line y = x – flips the graph of a function over the line y = x Symbolically (x,y) (y,x)
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Ex 1 Find an equation for the reflection of across each axis
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Tonight’s Assignment P. 139 – 140 Ex # 3-24 m. of 3
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Ex: Express h(x) so that it represents the graph of f(x) = x2 – 3 reflected over the x-axis? y-axis?
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Stretching & Shrinking
Complete the exploration on p. 136 First 8 min. No talking Last 8 min. you can discuss with a neighbor
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Stretches and Shrinks Let c be a positive real number. The following transformations result in stretches or shrinks of the graph of y = f(x). Horizontal stretches or shrinks y = f(x/c) a stretch by a factor of c if c > 1 a shrink by a factor of c if c < 1 Vertical stretches or shrinks y = cf(x) a stretch by a factor of c if c > 1
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a) f(x) = |x + 2| b) f(x) = x2 + x - 2
Ex 3 Transform the given function by (a) a vertical stretch by a factor of 2 and (b) horizontal shrink by a factor of 1/3. a) f(x) = |x + 2| b) f(x) = x2 + x - 2
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Combining Transformations
The order in which transformations are performed often affect the graph that results Ex 4 Use f(x) = x2 to perform each transformation. Write the formula for the resulting function. A horizontal shift 2 units to the right, a vertical stretch by a factor of 3, and vertical translation 5 units up Apply the transformations in the reverse order Are the graphs the same? Are the formulas the same?
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