Application of Functions Study Guide PDF
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Uploaded by ConciliatoryCarnelian8893
SEV American College
2024
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Summary
This document provides a comprehensive guide to the application of functions, including topics such as transformations, vertical and horizontal shifts, reflections, and function analysis. The material includes examples, exercises, and graphs to enhance understanding. This appears to be a study guide intended for high school students covering key concepts in algebra.
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2024-2025 ACADEMIC YEAR Name/Surname : Section, No : Topic: APPLICATION OF FUNCTIONS TRANSFORMATIONS Keywords : Transformation, Vertical Shifting, Horizontal Shi...
2024-2025 ACADEMIC YEAR Name/Surname : Section, No : Topic: APPLICATION OF FUNCTIONS TRANSFORMATIONS Keywords : Transformation, Vertical Shifting, Horizontal Shifting, Reflection, Stretching or Compressing, Rotation Vertical Shift: b>0, graph of 𝑓(𝑥) + 𝑏 shifts along the y-axis as follows. Example: 1 Example: Consider the function 𝑓(𝑥) = 𝑥 ! and sketch the graph of 𝑦 = 𝑥 ! + 3and 𝑦 = 𝑥 ! − 2, below. Activity 1: a) Give a name for the dotted function and try to write the solid one in terms of that function? b) If the dotted function above is f(x)+3 , write the solid one in terms of f(x). REMARK: For y=f(x)+b , the effect of b is to shift the graph vertically b units. If b>0 it moves upward If b 0 it moves to the right If a < 0 it moves to the left Activity 3: Give a name for the dotted function and try to write the solid one in terms of that function. Name of dotted function…………………………………………………… Solid function …………………………………………………………………. If the dotted function is f(x-2) , then write the solid function in terms of f. …………………………………………………………………………. Note: Remember, for horizontal shifts, it is opposite of what you see in the brackets. 3 EXERCISES 1) Describe the transformation of f(x) = (x+4)2 −5 using parent function 𝑦 = 𝑥 ! and sketch its graph. " 2) Given that g(x) =. Determine the equation of y= g(x-5)+3 and sketch its graph. # 4 3) Given h(x) = √𝑥. a) Use function notation to describe the graph of h(x), shifted left 4 units and up 2 units. b) Write the equation of the translated function described in part (a) and sketch its graph. 4) Draw the graph of 𝑦 = 𝑓(𝑥 − 2) − 3 on the same set of axis. a) b) 5 5) The graph of f(x) = x2 -2x+2 is translated 3 units right to g(x). Find g(x) in the form of g(x)=ax2+bx+c 6) Suppose f(x) = x2 is transformed to g(x) = (x-3)2+2. a) The coordinates of points on f(x) are given below. Find their transformed coordinates on g(x). i) (-3,9) ii) (0,0) Image:……………………. Image:…………………….. b) The coordinates of points on g(x) are given below. Find their transformed coordinates on f(x). i) (1,6) ii) (-2,27) Image:……………………. Image:…………………….. 6 Reflections Graphs of 𝑦 = 𝑓(𝑥) and 𝑦 = −𝑓(𝑥) are reflected about 𝑥-axis Graphs of 𝑦 = 𝑓(𝑥) and 𝑦 = 𝑓(−𝑥) are reflected about 𝑦-axis Example 1: If 𝑓(𝑥) = 3𝑥 + 2, state the equation of the function that is: a) reflection in the y-axis b) reflection in the x-axis Example 2: Graph of 𝑓(𝑥) is given below, on the same axes, graph 𝑓(−𝑥). 7 Example 3: For the graph of y= g(x) given, sketch the graph of: a) 𝑦 = 𝑔(𝑥) + 2 b) 𝑦 = −𝑔(𝑥) c) 𝑦 = 𝑔(−𝑥) d) 𝑦 = 𝑔(𝑥 + 1) a) b) c) d) 8 VERTICAL STRETCH and SHRINK: 𝑦 = 𝑓(𝑥) is a function and c is a positive real number. When 𝑐 > 1, the graph of 𝑦 = 𝑐. 𝑓(𝑥) is the graph of 𝑦 = 𝑓(𝑥) vertically stretched by multiplying each of its y-coordinates by c. When 0 < 𝑐 < 1, the graph of 𝑦 = 𝑐. 𝑓(𝑥) is the graph of 𝑦 = 𝑓(𝑥) vertically shrunk by multiplying each of its y-coordinates by c. vertically stretching (c >1) vertically shrinking (0