Grade 12 Mathematics Functions from a Calculus Perspective

Grade 12 Mathematics Functions from a Calculus Perspective ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 1. 3 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 1. 5 Transforming functions Earlier in the course we looked at the following transformations on a general function f(x): y = f(x) + a is a translation of a units in the y-direction. a y = f(x + a) is a translation of –a units in the x-direction. a y = –f(x) is a reflection in the x-axis. y = f(–x) is a reflection in the y-axis. y = af(x) is a one-way stretch by scale factor a in the y direction. y = f(ax) is a one-way stretch by scale factor the x direction. in ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation a

PA 1. 5 Transforming graphs of functions Graphs can be transformed by translating, reflecting, stretching or rotating them. The equation of the transformed graph will be related to the equation of the original graph. When investigating transformations it is most useful to express functions using function notation. For example, suppose we wish to investigate transformations of the function f(x) = x 2. The equation of the graph of y = x 2, can be written as y = f(x). ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 1. 5 Vertical translations Here is the graph of y = x 2, where y = f(x). This is the graph of y = f(x) + 1 y and this is the graph of y = f(x) + 4. What do you notice? This is the graph of y = f(x) – 3 x and this is the graph of y = f(x) – 7. What do you notice? The graph of y = f(x) + a is the graph of y = f(x) translated by the vector 0. a ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 1. 5 Horizontal translations Here is the graph of y = x 2 – 3, where y = f(x). This is the graph of y = f(x – 1), y and this is the graph of y = f(x – 4). What do you notice? This is the graph of y = f(x + 2), x and this is the graph of y = f(x + 3). What do you notice? The graph of y = f(x + a ) is the graph of y = f(x) translated by the vector –a. 0 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 1. 5 Reflections in the x-axis Here is the graph of y = x 2 – 2 x – 2, where y = f(x). y This is the graph of y = –f(x). What do you notice? x The graph of y = –f(x) is the graph of y = f(x) reflected in the x-axis. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 1. 5 Reflections in the y-axis Here is the graph of y = x 3 + 4 x 2 – 3 where y = f(x). y This is the graph of y = f(–x). What do you notice? x The graph of y = f(–x) is the graph of y = f(x) reflected in the y-axis. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 1. 5 Stretches in the y-direction Here is the graph of y = x 2, where y = f(x). This is the graph of y = 2 f(x). y What do you notice? This graph is is produced by doubling the y-coordinate of every point on the original graph y = f(x). This has the effect of stretching the graph in the vertical direction. x The graph of y = af(x) is the graph of y = f(x) stretched parallel to the y-axis by scale factor a. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 1. 5 Stretches in the x-direction Here is the graph of y = x 2 + 3 x – 4, where y = f(x). This is the graph of y = f(2 x). y What do you notice? x This graph is is produced by halving the x-coordinate of every point on the original graph y = f(x). This has the effect of compressing the graph in the horizontal direction. The graph of y = f(ax) is the graph of y = f(x) stretched parallel to the x-axis by scale factor 1 a. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 1. 5 Transforming linear functions ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 1. 5 Transforming quadratic functions ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 1. 5 Transforming cubic functions ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 1. 5 Transforming functions We can now look at what happens when we combine any of these transformations. For example, since all quadratic curves have the same basic shape any quadratic curve can be obtained by performing a series of transformations on the curve y = x 2. Write down the series of transformations that must be applied to the graph of y = x 2 to give the graph y = 2 x 2 + 4 x – 1. Start by writing y = 2 x 2 + 4 x – 1 in completed square form: 2 x 2 + 4 x – 1 = 2(x 2 + 2 x) – 1 = 2((x + 1)2 – 1) – 1 = 2(x + 1)2 – 3 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Transforming functions PA 1. 5 So, starting with y = x 2: y = x 2 1. Translate – 1 units in the x-direction. y = (x + 1)2 2. Stretch by a scale factor of 2 in the y-direction. y = 2(x + 1)2 3. Translate – 3 units in the y-direction. y = 2(x + 1)2 – 3 These transformations must be performed in the correct order. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 1. 5 Combining transformations 1 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 1. 5 Combining transformations 2 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 1. 5 Combining transformations 3 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 1. 5 Combining transformations 4 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

The graph of y = |x| PA 1. 5 Let’s start with the graph of y = x: y x| y = |x x For positive values of x the graph of y = |x| will be the same as the graph of y = x. For negative values of x the graph of y = |x| will be the same as the graph of y = –x. Notice that the part of y = x that lies below the x-axis is reflected in the x-axis to give y = |x|. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 1. 5 The graph of y = |f(x)| In general, the graph of y = f(x) can be sketched by reflecting above the x-axis the parts of y = f(x) that are below the x-axis. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 1. 5 The graph of y = f(|x|) Suppose we want to sketch the graph of y = |x|2 – |x| – 2. Start by sketching y = x 2 – x – 2. This can be factorized as y = (x + 1)(x – 2) and so it has roots at x = – 1 and x = 2: For positive values of x, the 2 2 x ––x |x| – 2– 2 y y = |x| graph of y = |x|2 – |x| – 2 will be the same as the graph of y = x 2 – x – 2. For negative values of x, the graph of y = |x|2 – |x| – 2 will – 2 – 1 2 be the same as the graph of y x = (–x)2 – (–x ) – 2. = x 2 + x – 2 = (x – 1)(x + 2) ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 1. 5 The graph of y = f(|x|) In general, for positive values of x (that is, for parts of the curve to the right of the y-axis) the graph of y = f(|x|) will be the same as the graph of y = f(x). For negative values of x (that is, for parts of the curve to the left of the y-axis) the graph of y = f(|x|) will be the same as the graph of y = f(–x). From previous work on transformations you will remember that the graph of y = f(–x) is a reflection of the graph of y = f(x) in the y-axis. Therefore, starting with the graph of y = f(x), the graph of y = f(|x|) can be sketched by reflecting left of the y-axis any part of y = f(x) that is to the right of the y-axis. Any part of y = f(x) that is to the left of the y-axis is ignored. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 1. 5 Left of the y-axis y = f(–x) The graph of y = f(|x|) Right of the y-axis y = f(x) ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation