Question

NO LINKS! Please help me with these problems​

Answer 1

4. Translated 6 units left and 1 unit down.

5. Translated 3 units right and stretched vertically by a factor of 2.

Step-by-step explanation:

Transformations

`\textsf{For $a > 0$}`

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

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

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

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

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

`f(ax) \implies f(x) \: \textsf{stretched parallel to the $x$-axis (horizontally) by a factor of $(1)/(a)$}`

`-f(x) \implies f(x) \: \textsf{reflected in the $x$-axis}`

`f(-x) \implies f(x) \: \textsf{reflected in the $y$-axis}`

Question 4

Parent function:

`f(x)=x^2`

Translated 6 units left:

Add 6 to the x-variable of the function:

`\implies f(x+6)=(x+6)^2`

Translated 1 unit down:

Subtract 1 from the function:

`\implies f(x+6)-1=(x+6)^2-1`

Question 5

Parent function:

`f(x)=|x|`

Translated 3 units right:

Subtract 3 from the x-variable of the function:

`\implies f(x-3)=|x-3|`

Stretched vertically by a factor of 2:

Multiply the function by 2:

`\implies 2f(x-3)=2|x-3|`

Answer 2

Answers in bold:

Problem 4) Shift 6 units left, 1 unit down

Problem 5) Shift 3 units right. Vertically stretch by a factor of 2.

The graphs are below.

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Step-by-step explanation:

In problem 4, the parent function is y = x^2

Replace every x with (x+6) and it will shift the parabola 6 units left. Why left instead of right? It's because the xy axis is moving 6 units to the right. Each old input x is now 6 units larger to get x+6. The xy axis moving 6 units right gives the illusion the curve shifts 6 units left. It's a bit backwards I know.

Luckily the -1 at the end is straight forward and it shifts everything down by 1 unit. This is because we subtract 1 from the y coordinate.

So that's how we get the "6 units left, 1 unit down" translation. No horizontal nor vertical dilations occur.

I recommend using GeoGebra or Desmos or similar to graph out the two equations, to see the comparison.

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For problem 5, the parent function is y = |x|

Replacing every x with x-3 means that the xy grid moves 3 units left. That gives the illusion the V shaped curve is moving 3 units right.

The 2 out front will double each y coordinate. This visually stretches the graph by a factor of 2. It is now twice as tall as it was before. This is equivalent to doing a horizontal compression (since the graph is more skinny now).