HSC Mathematics Advanced - Functions PDF

HSC MATHEMATICS ADVANCED FUNCTIONS Presented by Emma Crosswell © Emma Crosswell & Edrolo 2019 1 HSC MATHEMATICS ADVANCED – FUNCTIONS MA-F2: GRAPHING TECHNIQUES Presented by Emma Crosswell © Emma Crosswell & Edrolo 2019 2 HSC MATHEMATICS ADVANCED – FUNCTI...

HSC MATHEMATICS ADVANCED FUNCTIONS Presented by Emma Crosswell © Emma Crosswell & Edrolo 2019 1 HSC MATHEMATICS ADVANCED – FUNCTIONS MA-F2: GRAPHING TECHNIQUES Presented by Emma Crosswell © Emma Crosswell & Edrolo 2019 2 HSC MATHEMATICS ADVANCED – FUNCTIONS – MA-F2: GRAPHING TECHNIQUES TRANSFORMATIONS OF FUNCTIONS Presented by Emma Crosswell Syllabus dot point: apply transformations to sketch functions of the form 𝑦 = 𝑘𝑓(𝑎(𝑥 +𝑏))+𝑐, where 𝑓(𝑥) is a polynomial, reciprocal, absolute value, exponential or logarithmic function and 𝑎, 𝑏, 𝑐 and 𝑘 are constants – examine translations and the graphs of 𝑦 = 𝑓(𝑥)+𝑐 and 𝑦 = 𝑓(𝑥 +𝑏) using technology – examine dilations and the graphs of 𝑦 = 𝑘𝑓(𝑥) and 𝑦 = 𝑓(𝑎𝑥) using technology – recognise that the order in which transformations are applied is important in the construction of the resulting function or graph © Emma Crosswell & Edrolo 2019 3 What’s in this lesson? Vertical and horizontal translations Vertical and horizontal dilations Combination of transformations © Emma Crosswell & Edrolo 2019 4 Transformations What’s in this lesson? review game quizizz.com/admin/q Reflection Rotation R uiz/56d8ccb2264bfc b94d8896a3/trans formations-review R Translation Dilation T A) T B) A B C C) © Emma Crosswell & Edrolo 2019 5 Function (n.) Vertical and horizontal translations A special type of Consider the following video which shows the graph of 𝑦 = 𝑥 ) + 𝑐, where 𝑐 is a parameter. relation where for every value of 𝑥, there is exactly one value of 𝑦. Translation (n.) A translation moves an object without changing its shape. A function 𝑦 = 𝑓 𝑥 + 𝑐 translates the graph of the function 𝑓(𝑥) vertically. If 𝑐 > 0, the graph is translated up by 𝑐 units. If 𝑐 < 0, the graph is translated down by 𝑐 units. Graph image source: Desmos © Emma Crosswell & Edrolo 2019 6 Function (n.) Vertical and horizontal translations A special type of Consider the following video which shows the graph of 𝑦 = 𝑥 + 𝑏 ) , where 𝑏 is a parameter. relation where for every value of 𝑥, there is exactly one value of 𝑦. Translation (n.) A translation moves an object without changing its shape. A function 𝑦 = 𝑓(𝑥 + 𝑏) translates the graph of the function 𝑓(𝑥) horizontally. If 𝑏 > 0, the graph is translated left by 𝑏 units. If 𝑏 < 0, the graph is translated right by 𝑏 units. Graph image source: Desmos © Emma Crosswell & Edrolo 2019 7 Worked example Explain how the graphs of 𝑦 = 𝑥 ) − 5 and 𝑦 = 𝑥 − 5 ) are related to the graph of 𝑦 = 𝑥 ). © Emma Crosswell & Edrolo 2019 8 Image (n.) Worked example The transformed The point 𝑄(−4, −3) lies on a function 𝑦 = 𝑔(𝑥). Find the image point if the function is translated coordinate, graph or 6 units up. equation. © Emma Crosswell & Edrolo 2019 9 Worked example The following is the graph of 𝑦 = |𝑥|. Sketch the graphs of 𝑦 = 𝑥 − 4 and 𝑦 = |𝑥 + 4|. Graph image source: Desmos © Emma Crosswell & Edrolo 2019 10 Multiple choice activity What is the equation of the function 𝑓 𝑥 = 𝑥 ; − 3𝑥 ) + 2𝑥 − 1 translated 3 units up? A. 𝑓 𝑥 = 𝑥 ; − 3𝑥 ) + 2𝑥 − 1 B. 𝑓 𝑥 = 𝑥 ; − 3𝑥 ) + 2𝑥 − 3 C. 𝑓 𝑥 = 𝑥 ; − 3𝑥 ) + 2𝑥 + 2 D. 𝑓 𝑥 = 𝑥 ; − 3𝑥 ) + 2𝑥 + 3 E. I don’t know. © Emma Crosswell & Edrolo 2019 (Written by the author) 11 Multiple choice – Response What is the equation of the function 𝑓 𝑥 = 𝑥 ; − 3𝑥 ) + 2𝑥 − 1 translated 3 units up? A. 𝑓 𝑥 = 𝑥 ; − 3𝑥 ) + 2𝑥 − 1 B. 𝑓 𝑥 = 𝑥 ; − 3𝑥 ) + 2𝑥 − 3 C. 𝒇 𝒙 = 𝒙𝟑 − 𝟑𝒙𝟐 + 𝟐𝒙 + 𝟐 D. 𝑓 𝑥 = 𝑥 ; − 3𝑥 ) + 2𝑥 + 3 E. I don’t know. © Emma Crosswell & Edrolo 2019 (Written by the author) 12 Multiple choice activity A point 𝑃(3, −5) lies on the function 𝑦 = 𝑓(𝑥). What is the image point if the function is transformed to 𝑦 = 𝑓(𝑥 + 𝑏) where 𝑏 = −4? A. −1, −5 B. 7, −5 C. 3, −9 D. (3, −1) E. I don’t know. © Emma Crosswell & Edrolo 2019 (Written by the author) 13 Multiple choice – Response A point 𝑃(3, −5) lies on the function 𝑦 = 𝑓(𝑥). What is the image point if the function is transformed to 𝑦 = 𝑓(𝑥 + 𝑏) where 𝑏 = −4? A. −1, −5 B. 𝟕, −𝟓 C. 3, −9 D. (3, −1) E. I don’t know. © Emma Crosswell & Edrolo 2019 (Written by the author) 14 Dilation (n.) Vertical and horizontal dilations A dilation stretches Consider the following video which shows the graph of 𝑦 = 𝑘(𝑥 ; + 𝑥 ) − 𝑥), where 𝑘 is a parameter. or compresses a function and changes its size and shape. A function 𝑦 = 𝑘𝑓(𝑥) dilates the graph of the function 𝑓(𝑥) vertically by a scale factor of 𝑘. If 𝑘 > 1, the graph is stretched. If 0 < 𝑘 < 1, the graph is compressed. If 𝑘 = −1, the graph reflected in the 𝑥-axis. Graph image source: Desmos © Emma Crosswell & Edrolo 2019 15 Vertical and horizontal dilations I Consider the following video which shows the graph of 𝑦 = 𝑓(𝑎𝑥) where 𝑓 𝑥 = ; 𝑥 ; − 𝑥 ) − 24𝑥, where 𝑎 is a parameter. I A function 𝑦 = 𝑓(𝑎𝑥) dilates the graph of the function 𝑓(𝑥) horizontally by a scale factor of. J If 𝑎 > 1, the graph is compressed. If 0 < 𝑎 < 1, the graph is stretched. If 𝑎 = −1, the graph reflected in the 𝑦-axis. Graph image source: Desmos © Emma Crosswell & Edrolo 2019 16 Worked example Explain how the graphs of 𝑦 = 6 log 𝑥 and 𝑦 = log(6𝑥) are related to the graph of 𝑦 = log 𝑥. © Emma Crosswell & Edrolo 2019 17 Worked example Find the equation of the transformed graph if 𝑦 = |𝑥| is reflected in the 𝑥-axis and dilated vertically by a factor of 3. © Emma Crosswell & Edrolo 2019 18 Multiple choice activity What is the equation of the function 𝑓 𝑥 = 𝑥 O − 3𝑥 + 1 reflected in the 𝑦-axis? A. 𝑓 𝑥 = −𝑥 O + 3𝑥 − 1 B. 𝑓 𝑥 = −𝑥 O − 3𝑥 + 1 C. 𝑓 𝑥 = 𝑥 O − 3𝑥 + 1 D. 𝑓 𝑥 = 𝑥 O + 3𝑥 + 1 E. I don’t know. © Emma Crosswell & Edrolo 2019 (Written by the author) 19 Multiple choice – Response What is the equation of the function 𝑓 𝑥 = 𝑥 O − 3𝑥 + 1 reflected in the 𝑦-axis? A. 𝑓 𝑥 = −𝑥 O + 3𝑥 − 1 B. 𝑓 𝑥 = −𝑥 O − 3𝑥 + 1 C. 𝑓 𝑥 = 𝑥 O − 3𝑥 + 1 D. 𝒇 𝒙 = 𝒙𝟒 + 𝟑𝒙 + 𝟏 E. I don’t know. © Emma Crosswell & Edrolo 2019 (Written by the author) 20 Multiple choice activity The graph of 𝑦 = 𝑓(𝑥) is shown in red. Which of the following is a possible equation of the second graph in purple? 𝒚= A. 𝑦 = 𝑓(3𝑥) B. 𝑦 = 3𝑓(𝑥) I C. 𝑦 = 𝑓 ; 𝑥 𝒚 = 𝒇(𝒙) I D. 𝑦 = ; 𝑓 𝑥 E. I don’t know. Graph image source: Desmos © Emma Crosswell & Edrolo 2019 (Written by the author) 21 Multiple choice – Response The graph of 𝑦 = 𝑓(𝑥) is shown in red. Which of the following is a possible equation of the second graph in purple? 𝒚= A. 𝒚 = 𝒇(𝟑𝒙) B. 𝑦 = 3𝑓(𝑥) I C. 𝑦 = 𝑓 ; 𝑥 𝒚 = 𝒇(𝒙) I D. 𝑦 = ; 𝑓 𝑥 E. I don’t know. Graph image source: Desmos © Emma Crosswell & Edrolo 2019 (Written by the author) 22 Transformations Combination of transformations (n.) A function can have a number of different transformations acting on it. The process of changing a function The function 𝑦 = 𝑘𝑓 𝑎 𝑥 + 𝑏 + 𝑐 is a transformation of 𝑦 = 𝑓 𝑥 with a: through dilations and translations. if 𝑐 > 0, the graph shifts up Vertical translation of 𝑐 where if 𝑐 < 0, the graph shifts down if 𝑏 > 0, the graph shifts to the left Horizontal translation of 𝑏 where if 𝑏 < 0, the graph shifts to the right if 𝑘 > 1, the graph is stretched vertically with scale factor 𝑘 Vertical dilation of 𝑘 where if 0 < 𝑘 < 1, the graph is compressed vertically with scale factor 𝑘 if 𝑘 = −1, the graph is reflected in the 𝑥-axis I if 𝑎 > 1, the graph is compressed horizontally with scale factor J Horizontal dilation of 𝑎 where I if 0 < 𝑎 < 1, the graph is stretched horizontally with scale factor J if 𝑎 = −1, the graph is reflected in the 𝑦-axis Order of transformations: 1. Dilations (including reflections) 2. Translations © Emma Crosswell & Edrolo 2019 23 Worked example Find the equation of the transformed function of 𝑦 = log 𝑥 which has been translated down by 4 units, translated 3 units to the left and then is vertically dilated with scale factor 6. © Emma Crosswell & Edrolo 2019 24 Worked example The function 𝑦 = 𝑓(𝑥) has a stationary point at (2, −1). It is transformed to 𝑦 = 4𝑓 2(𝑥 − 1) + 3. Where is the new stationary point? © Emma Crosswell & Edrolo 2019 25 Worked example I ; The following is the graph of 𝑦 =. Sketch the graph of 𝑦 = 5 −. S ST) Graph image source: Desmos © Emma Crosswell & Edrolo 2019 26 Multiple choice activity What is the equation of the the transformed function if 𝑓 𝑥 = 𝑥 has a vertical translation of I 3 units down, a horizontal translation of 6 to the left and a horizontal dilation with scale factor )? A. 𝑓 𝑥 = 2(𝑥 − 6) − 3 I B. 𝑓 𝑥 = ) (𝑥 − 6) − 3 C. 𝑓 𝑥 = 2(𝑥 + 6) − 3 I D. 𝑓 𝑥 = ) (𝑥 + 6) − 3 E. I don’t know. © Emma Crosswell & Edrolo 2019 (Written by the author) 27 Multiple choice – Response What is the equation of the the transformed function if 𝑓 𝑥 = 𝑥 has a vertical translation of I 3 units down, a horizontal translation of 6 to the left and a horizontal dilation with scale factor )? A. 𝑓 𝑥 = 2(𝑥 − 6) − 3 I B. 𝑓 𝑥 = ) (𝑥 − 6) − 3 C. 𝒇 𝒙 = 𝟐(𝒙 + 𝟔) − 𝟑 I D. 𝑓 𝑥 = ) (𝑥 + 6) − 3 E. I don’t know. © Emma Crosswell & Edrolo 2019 (Written by the author) 28 Multiple choice activity The function 𝑦 = |𝑥| passes through the point (1, 1). What are the coordinates of the image point when the function is transformed into 𝑦 = −3 2𝑥 − 1 − 1? A. (1, 2) B. (1, −4) V C. ) , −4 D. (3, −4) E. I don’t know. © Emma Crosswell & Edrolo 2019 (Written by the author) 29 Multiple choice – Response The function 𝑦 = |𝑥| passes through the point (1, 1). What are the coordinates of the image point when the function is transformed into 𝑦 = −3 2𝑥 − 1 − 1? A. (1, 2) B. (𝟏, −𝟒) V C. ) , −4 D. (3, −4) E. I don’t know. © Emma Crosswell & Edrolo 2019 (Written by the author) 30  The copyright in substantial portions of this material is owned by the NSW Education Standards Authority. Used with permission. The NESA does not endorse this product and makes no warranties regarding the correctness or accuracy of its content. To the extent permitted by law, the NESA excludes all liability for any loss or damage suffered or incurred as a result of accessing, using or relying on the content. Current and past NESA/BOSTES exams and related content can be accessed directly at http://educationstandards.nsw.edu.au. We do our best to make these slides comprehensive and up-to-date, however there may be errors. We'd appreciate it if you pointed these out to us! © Emma Crosswell & Edrolo 2019 31