Combinations of Transformations MCR3U Stretched vertically by a factor of đ ï· If |đ| > 1, the graph is expanded ï· If 0 < |đ| < 1 the graph is compressed ï· If đ < 0, the graph is reflected in the đ„-axis Translated vertically đ units y ïœ af ïš k ïš x ï d ï© ï© ï« c Stretched horizontally by a factor of đ/đ ï· If |đ| > 1, the graph is compressed ï· If 0 < |đ| < 1 the graph is expanded ï· If đ < 0, the graph is graph is reflected in the đŠ-axis ï· If đ > 0, the graph shifts up ï· If đ < 0, the graph shifts down Translated horizontally đ units ï· If the sign is negative, the graph shifts to the right ï· If the sign is positive, the graph shifts to the left The đ affects the graph y ïœ f ïš x ï© by stretching or compressing ___________________ by a factor of đ. If the đ is negative, there is a reflection about the __________________. The đ affects the graph y ïœ f ïš x ï© by translating ______________________ đ units. The đ affects the graph y ïœ f ïš x ï© by translating ______________________ đ units. The đ affects the graph y ïœ f ïš x ï© by stretching or compressing ____________________ by a factor of 1 . k If the đ is negative, there is a reflection about the _________________. Does the order of transformations matter? Graphing Point by Point 1) Start by graphing the base function ( y ïœ x 2 , y ïœ x , y ïœ x3 , y ïœ x , y ïœ 1 ). x 2) Graph đ by taking the significant point (đ„, đŠ) and multiply only the đŠ-values by đ so that the point (đ„, đŠ) becomes (đ„, đđŠ). 3) Graph đ by taking the significant point (đ„, đŠ) and multiply only the đ„-values by 1 so that the k ïŠx ï¶ point (đ„, đŠ) becomes ï§ , y ï· . ïšk ïž 4) Graph đ by taking the significant point (đ„, đŠ) and adding đ to the đ„-value so that the point (đ„, đŠ) becomes (đ„ + đ, đŠ). 5) Graph đ by taking the significant point (đ„, đŠ) and adding đ to the đŠ-value so that the point (đ„, đŠ) becomes (đ„, đŠ + đ). Example 1 – Sketching Graphs of Transformed Functions 1. Given the graph of đ(đ„) = |đ„|, describe how you would graph đ(đ„) = −đ(đ„ − 3) + 4 using transformations. 2. a) Using transformations, sketch the graph of đŠ = √2đ„ + 4 + 1. Hint: Rewrite 2đ„ + 4 in factored form to determine the horizontal translation. Example 2 – Writing Equations of Transformed Functions 1. The function đŠ = đ(đ„) has been transformed into đŠ = đđ(đ(đ„ − đ)) + đ. Write the following in the appropriate form: 1 (a) a vertical compression by a factor of 2 , a reflection in the đ„-axis and a translation 3 units right. (b) a vertical stretch by a factor of 3, a horizontal stretch by a factor of 2, a translation left 5 and up 4, and a reflection in the đŠ-axis. Practice Transformations Given an Equation Graph each of the following functions by: 1ï¶ ïŠ a) Graphing the base function first. ï§ y ïœ x 2 , y ïœ x , y ïœ x3 , y ïœ x , y ïœ ï· xïž ïš b) Listing the transformations. c) Applying the transformations to the base function. 1) y ïœ 2 ïš x ï« 1ï© ï 1 2 2) y ïœ 2 ï«1 xï«2 3) y ïœ ïš 2 x ï 2 ï© ï 1 3 4) y ïœ ï x ï 2 ï« 1 7) y ïœ 1 1 xï 2 2 6) y ïœ 3 x ï« 1 ï« 1 5) y ïœ 2 3 x ïŠ 1 ï¶ 8) y ïœ ï ï§ ï ïš x ï« 1ï© ï· ïš 2 ïž 3 9) y ïœ 2 2 x ï« 2 ï« 2 Practice Transformations Given a Graph List the transformations. Apply the transformations to key points on the graph. 1) y ïœ 3g ïš ï2 ïš x ï 1ï© ï© ï« 1 3) y ïœ 1 h ïš x ï« 1ï© ï 2 2 2) y ïœ ï f ïš 2 ïš x ï« 1ï© ï© 4) y ïœ ï 1 f ïš 2x ï« 4ï© ï1 2 5) y ïœ f (3x ï 6) 6) y ïœ f (ï2 x ï« 4) 7) y ïœ ï2 f ( x ï 3) ï« 1 8) y ïœ 1 ïŠ1 ï¶ f ï§ x ï 2ï· 2 ïš3 ïž