Question

HELPPPP PLSSSSSSSS -----

Answer 1

Answer:

  a) vertical expansion by a factor of 2; translation 3 units left and 5 units down

  b) see attached

Step-by-step explanation:

a.

Describing transformations is all about matching patterns. The elements of the transformed function are matched with the elements of a transformation.

Vertical scaling

A function is scaled vertically by multiplying each function value by some scale factor. In generic terms, the function f(x) is scaled vertically by the factor 'c' in this way:

• original function: f(x)
• scaled by a factor of 'c': c·f(x)

If we want the function f(x) = x² scaled vertically by a factor of 2, then we have

  f(x) = x² . . . . . . . original function

  2·f(x) = 2x² . . . . scaled vertically by a factor of 2

On a graph, each point is vertically twice as far vertically from some reference point (the vertex, for example) as it is in the original function graph.

Horizontal translation

A function is translated to the right by 'h' units when x is replaced by (x -h).

• original function: f(x)
• translated h units right: f(x -h)

If we want the function f(x) = x² translated right by 3 units, we will have ...

  f(x) = x² . . . . . . . . . . . original function

  f(x -3) = (x -3)² . . . . . .translated right 3 units

Note that translation left by 3 units would give ...

  f(x -(-3)) = f(x +3) = (x +3)² . . .  . translated left 3 units

On a graph, each point of the left-translated function is 3 units left of where it was on the original function graph.

Vertical translation

A function is translated upward by 'k' units when k is added to the function value.

• original function: f(x)
• translated k units up: f(x) +k

The value of k will be negative for a translation downward.

If we want the function f(x) = x² translated down by 5 units, we will have ...

  f(x) = x² . . . . . . . . . . . original function

  f(x) = x² -5 . . . . . . . . .translated down 5 units

Combined transformations

Using all of these transformations at once, we have ...

  f(x) = x² . . . . . . . . . . . . . . . . . original function

  c·f(x -h) +k = c·(x -h)² +k . . . scaled by 'c', translated h right and k up

Compare this to the given function:

 y = 2(x +3)² -5

and we can see that ...

• c = 2 . . . . . . vertical scaling by a factor of 2
• h = -3 . . . . . translation 3 units left
• k = -5 . . . . . translation 5 units down

This is the pattern matching that is described at the beginning.

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b.

When graphing a transformed function, it is often useful to start with a distinctive feature and work from there. The vertex of a parabola is one such distinctive feature.

Translation

The transformations move the vertex 3 units left and 5 units down from its original position at (0, 0). The location of the vertex on the transformed function graph will be at (x, y) = (-3, -5).

Vertical scaling

The graph of the parent function parabola (y= x²) goes up from the vertex by the square of the number of units right or left. That is, 1 unit right or left of the vertex, the graph is 1 unit above the vertex. 2 units right or left, the graph is 2² = 4 units above the vertex.

The scaled graph will have these vertical distances multiplied by 2:

• ±1 unit horizontally ⇒ 2·1² = 2 units vertically; points (-4, -3), (-2, -3)
• ±2 units horizontally ⇒ 2·2² = 8 units vertically; points (-5, 3), (-1, 3)

The graph of the transformed function is shown in blue in the attachment.

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Additional comment

The vertical scale factor 'c' may have any non-zero value, positive or negative, greater than 1 or less than 1. When the magnitude is less than 1, the scaling is a compression, rather than an expansion. When the sign is negative, the graph is also reflected across the x-axis, before everything else.

Answer 2

Answer:

Translation of 3 units to the left.

Vertical stretch by a factor of 2.

Translation of 5 units down.

Step-by-step explanation:

Transformations

For a > 0

$f(x+a) \implies f(x) \: \textsf{translated}\:a\:\textsf{units left}$

$y=a\:f(x) \implies f(x) \: \textsf{stretched parallel to the y-axis (vertically) by a factor of}\:a$

$f(x)-a \implies f(x) \: \textsf{translated}\:a\:\textsf{units down}$

Parent function:

$y=x^2$

Translate 3 units left

Add 3 to the variable of the function

$\implies y=(x+3)^2$

Stretch vertically by a factor of 2

Multiply the whole function by 2:

$\implies y=2(x+3)^2$

Translate 5 units down

Subtract 5 from the whole function:

$\implies y=2(x+3)^2-5$

Please see the attached graphs for the final transformed function (as well as the graphed steps).