Respuesta :

First of all, you have to know that the absolute function

[tex]f(x)=|x|[/tex]

is represented by the V-shaped graph, centered at the origin (see attached picture).

Once this is clear, it's all a matter of function transformation: for the first function you have:

[tex]|x|\mapsto |x-5|[/tex]

so, a transformation of the form

[tex]f(x)\mapsto f(x-5)[/tex]

this results in a horizontal translation, 5 units to the right (because you're subtracting 5 from the input).

Then, you multiply this by 3:

[tex]|x-5|\mapsto 3|x-5|[/tex]

so, a transformation of the form

[tex]f(x)\mapsto 3f(x)[/tex]

this results in a vertical stretch, with a scale factor 3 (because you're multiplying the whole function by 3).

Finally, you add 2, which results in a vetical translation, 2 units up. See the second attached picture for a comparison. As you can see, the child function is narrower (effect of the multiplication by 3), translated right by 5 and down by 2 units (effect of, respectively, subtracting 5 inside the absolute value and adding 2 to the whole function).

The second and third exercise follow the same logic, these are the steps:

[tex]|x|\mapsto |x+4|[/tex]

This is a horizontal translation, 4 units to the left

[tex]|x+4|\mapsto -\dfrac{1}{2}|x+4|[/tex]

This is a vertical stretch, scale factor 1/2, and reflection across the x axis (you're multiplying by a negative number).

[tex]-\dfrac{1}{2}|x+4|\mapsto -\dfrac{1}{2}|x+4|-3[/tex]

This is a vertical translation, 3 units down.

Perform these 3 transformations (in this order!) and you'll have the child function.

Finally, for the third function, we have

[tex]|x|\mapsto |x-3|[/tex]

This is a horizontal translation, 3 units to the right

[tex]|x-3|\mapsto -4|x-3|[/tex]

This is a vertical stretch, scale factor 4, and reflection across the x axis (you're multiplying by a negative number).

Perform these 2 transformations (in this order!) and you'll have the child function.

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tramserran

NOTES: y = a|x - h| + k

Vertex = (h, k)

AOS: x = h

Domain: x = All Real Numbers (-∞, ∞)

Range: if "a" is positive → y ≥ k

            if "a" is negative → y ≤ k

Comparison to Parent Function: "a" is vertical stretch

                                                      "h" is horizontal shift (left, right)

                                                       "k" is vertical shift (up, down)

Graph: Plot the vertex

Plot a point to the right by counting rise over run using the "a"-value

Plot a point to the left by counting rise over run using opposite of "a"-value

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4. f(x) = 3|x - 5| + 2

   a = 3     h = 5      k = 2

   Vertex: (5, 2)      AOS: x = 3

   Domain: x = ARN (-∞, ∞)   Range: y ≥ 2

   Comparison to Parent Function:

  • vertical stretch by a factor of 3
  • horizontal shift 5 units to the right
  • vertical shift 2 units up

   Graph: see attachment 1

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[tex]5.\ f(x)=-\dfrac{1}{2}|x + 4| - 3[/tex]

   a = [tex]\underline{\ -\dfrac{1}{2}\ }[/tex]     h = -4      k = -3

   Vertex: (-4, -3)      AOS: x = -4

   Domain: x = ARN (-∞, ∞)   Range: y ≤ -3

   Comparison to Parent Function:

  • reflection over the x-axis
  • vertical shrink by a factor of 1/2
  • horizontal shift 4 units to the left
  • vertical shift 3 units down

   Graph: see attachment 2

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6. f(x) = -4|x - 3|

   a = -4     h = 3      k = 0

   Vertex: (3, 0)      AOS: x = 3

   Domain: x = ARN (-∞, ∞)   Range: y ≤ 0

   Comparison to Parent Function:

  • reflection over the x-axis
  • vertical stretch by a factor of 4
  • horizontal shift 3 units to the right

   Graph: see attachment 3

Ver imagen tramserran
Ver imagen tramserran
Ver imagen tramserran