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HPC 4.4 – Horizontal Transformations Now suppose we want to dilate (stretch/compress) the squaring function by a factor of 3, meaning to stretch the graph three times as wide. 1. Draw the dilated function on the graph at the right using a red pen. Include on your graph the new location of the original 5 key points displayed on the parent graph. 2. Give the point rule: ( x, y ) _____________________ But how can we get an equation for the transformed graph? First, let’s consider the point rule relationship. We take a parent’ input (let’s call it xP ), multiply by three, and get the transformed input, xT . Symbolically, we can say xP g3 xT . Since the squaring relationship, output = (parent input)2, requires a parent input, let’s re-write the point-rule relationship: xP g3 xT as xP xT . This gives an expression for 3 the parent input based on the transformed x-value. 2 2 x x Thus, y ( parent input ) T . 3 3 2 3. Graph it on your calculator. Does the transformation stretch (dilate) the graph by a factor of 3? Determining the Transformation from the Equation 2 3x 4. You are given the equation: y . 2 a. What is the parent function? b. Determine the point-rule for the transformation on the parent, recalling that the x in the given function is really xT and you are trying to get an expression to represent xP . c. What transformation was done on the parent function? Be to include the words horizontal or vertical. sure d. Draw a sketch of the transformed function based on parts a b on the graph at the right. Include the transformed points your graph. and on HPC 4.4 – Horizontal Transformations Putting It All Together Now we need to get comfortable with notation. When given a function that has had a horizontal translation, horizontal dilation, vertical translation and vertical dilation, it is very important identify the parent function and then to decide what values are affecting the inputs and what values are affecting the outputs and in what order. 2 Example, suppose f ( x) 3 2 x 10 4 . Describe the transformations made on y x . 2 First, identify the parent function, which is y x . Then re-write the function using a general function equation: y 3g From this, the parent input expression is 2x+6, and the output expression is 3y + 4. Isolate the transformed input: 2 xT 10 xP leads to xT 2 Remember, y f ( x) x means yP f ( xP ) xP , so we easily determine yT 3 yP 4 . Therefore, the point-rule is: ( x, y) Now we can describe the transformations made on y x : 2 f (2 x 10) 4 . xP 10 . 2 2 x 10 , 1 2 3 y 4 . 2 shift left 10 units, horizontal dilation by a factor of ½, vertical dilation by a factor of 3, and shift up 4 units. HPC 4.4 – Horizontal Transformations A. PRACTICE WITH THE CONCEPT 1. Describe the transformation(s) on the following functions. Include a point-rule. b. b( x) cos(4 x) c. a. a ( x ) x 5 d. d ( x) g. 1 careful 3x g ( x) sin 2( x 3) e. e( x) sin x 1 h. h( x) cos(5 x 10) c( x) x 2 f. x f ( x) log 2 i. i ( x) 2 4 x 8 2. The point (3, 10) is on the graph of y = f(x). Find the coordinates of the corresponding point on the graph of: a) y = 5f(x-2) b) y = f(5x-2) c) y = f(5x) -2 d) y = f(0.5x-5) e) y = 0.5f(x+5) f) y = 0.5f(x)+5 g) y = 2f(x-3) h) y = 5 - f(2x) i) y = 2 + 3f(x) HPC 4.4 – Horizontal Transformations 3. Write the equation of the reflection of y sin(2 x 6) 1 a) in the x –axis b) in the origin c) in the y-axis 4. Describe in words how each transformation shifts, reflects, dilates the graph of a given function. a) y = f(x + 3) for y cos x b) y = f(x) + 3 for y csc x c) y = f(2x) for y tan x d) y = 2f(x) for y sin( x) e) y = f(-x) for y sin x f) y = -f(x) for y cos x g) y = f(2x - 6) for y cos x h) y = 2f(x) - 6 for y sin x 5. On the axis below, a. you are given y = g(x). Sketch y = g(x+2) - 3. c. you are given y = f(x). Sketch y = f(3x+6). b. you are given y = h(x). Sketch y = -h(2x).