exercises. describe the translation in each function. then graph the function. 1. y = x - 4 2. y = |x + 5|…

exercises. describe the translation in each function. then graph the function. 1. y = x - 4 2. y = |x + 5| 3. y = x^2 - 3. describe the dilation in each function. then graph the function. 4. y = 5x 5. y = 1/2|x| 6. y = 2x^2

exercises. describe the translation in each function. then graph the function. 1. y = x - 4 2. y = |x + 5| 3. y = x^2 - 3. describe the dilation in each function. then graph the function. 4. y = 5x 5. y = 1/2|x| 6. y = 2x^2

Answer

  1. For (y = x - 4):
    • Explanation:
      • Step 1: Identify the type of transformation
        • The parent - function is (y = x). The function (y=x - 4) is of the form (y=f(x)-k) where (f(x)=x) and (k = 4).
        • This represents a vertical translation.
      • Step 2: Determine the direction and distance of translation
        • For a function (y=f(x)-k) ((k>0)), the graph of (y = f(x)) is translated (k) units down. Here, the graph of (y = x) is translated 4 units down.
        • To graph (y=x - 4), we can find two points on the line. When (x = 0), (y=0 - 4=-4). When (y = 0), (0=x - 4), so (x = 4). Plot the points ((0,-4)) and ((4,0)) and draw a straight - line through them.
    • Answer: The graph of (y = x) is translated 4 units down.
  2. For (y=\vert x + 5\vert):
    • Explanation:
      • Step 1: Identify the type of transformation
        • The parent - function is (y=\vert x\vert). The function (y=\vert x + 5\vert) is of the form (y = f(x + h)) where (f(x)=\vert x\vert) and (h = 5).
        • This represents a horizontal translation.
      • Step 2: Determine the direction and distance of translation
        • For a function (y=f(x + h)) ((h>0)), the graph of (y = f(x)) is translated (h) units to the left. Here, the graph of (y=\vert x\vert) is translated 5 units to the left.
        • To graph (y=\vert x + 5\vert), we can consider cases. When (x+5\geq0) (i.e., (x\geq - 5)), (y=x + 5). When (x+5<0) (i.e., (x<-5)), (y=-(x + 5)=-x - 5). Plot key points such as when (x=-5), (y = 0); when (x=-4), (y = 1); when (x=-6), (y = 1) and draw the V - shaped graph.
    • Answer: The graph of (y=\vert x\vert) is translated 5 units to the left.
  3. For (y=x^{2}-3):
    • Explanation:
      • Step 1: Identify the type of transformation
        • The parent - function is (y=x^{2}). The function (y=x^{2}-3) is of the form (y=f(x)-k) where (f(x)=x^{2}) and (k = 3).
        • This represents a vertical translation.
      • Step 2: Determine the direction and distance of translation
        • For a function (y=f(x)-k) ((k>0)), the graph of (y = f(x)) is translated (k) units down. Here, the graph of (y=x^{2}) is translated 3 units down.
        • To graph (y=x^{2}-3), we can find some points. When (x = 0), (y=-3). When (x = 1), (y=1^{2}-3=-2). When (x=-1), (y=(-1)^{2}-3=-2). Plot the points and draw the parabola.
    • Answer: The graph of (y=x^{2}) is translated 3 units down.
  4. For (y = 5x):
    • Explanation:
      • Step 1: Identify the type of transformation
        • The parent - function is (y = x). The function (y = 5x) is of the form (y=af(x)) where (a = 5) and (f(x)=x).
        • This represents a vertical dilation.
      • Step 2: Determine the scale factor of dilation
        • For a function (y=af(x)) ((a>1)), the graph of (y = f(x)) is vertically dilated by a factor of (a). Here, the graph of (y = x) is vertically dilated by a factor of 5.
        • To graph (y = 5x), we can find two points. When (x = 0), (y = 0). When (x = 1), (y = 5). Plot the points ((0,0)) and ((1,5)) and draw a straight - line through them.
    • Answer: The graph of (y = x) is vertically dilated by a factor of 5.
  5. For (y=\frac{1}{2}\vert x\vert):
    • Explanation:
      • Step 1: Identify the type of transformation
        • The parent - function is (y=\vert x\vert). The function (y=\frac{1}{2}\vert x\vert) is of the form (y=af(x)) where (a=\frac{1}{2}) and (f(x)=\vert x\vert).
        • This represents a vertical dilation.
      • Step 2: Determine the scale factor of dilation
        • For a function (y=af(x)) ((0 < a<1)), the graph of (y = f(x)) is vertically compressed by a factor of (a). Here, the graph of (y=\vert x\vert) is vertically compressed by a factor of (\frac{1}{2}).
        • To graph (y=\frac{1}{2}\vert x\vert), consider cases. When (x\geq0), (y=\frac{1}{2}x). When (x<0), (y=-\frac{1}{2}x). Plot key points such as when (x = 0), (y = 0); when (x = 2), (y = 1); when (x=-2), (y = 1) and draw the V - shaped graph.
    • Answer: The graph of (y=\vert x\vert) is vertically compressed by a factor of (\frac{1}{2}).
  6. For (y = 2x^{2}):
    • Explanation:
      • Step 1: Identify the type of transformation
        • The parent - function is (y=x^{2}). The function (y = 2x^{2}) is of the form (y=af(x)) where (a = 2) and (f(x)=x^{2}).
        • This represents a vertical dilation.
      • Step 2: Determine the scale factor of dilation
        • For a function (y=af(x)) ((a>1)), the graph of (y = f(x)) is vertically dilated by a factor of (a). Here, the graph of (y=x^{2}) is vertically dilated by a factor of 2.
        • To graph (y = 2x^{2}), we can find some points. When (x = 0), (y = 0). When (x = 1), (y=2\times1^{2}=2). When (x=-1), (y=2\times(-1)^{2}=2). Plot the points and draw the parabola.
    • Answer: The graph of (y=x^{2}) is vertically dilated by a factor of 2.