Question

Please help with the attached question. Thanks

Answer 1

Choice A)
`F(x) = 3√(x + 1)`.

Explanation:

What are the changes that would bring
`G(x)` to
`F(x)`?

• Translate
  `G(x)` to the left by
  `1` unit, and
• Stretch
  `G(x)` vertically (by a factor greater than
  `1`.)

`G(x) = √(x)`. The choices of
`F(x)` listed here are related to
`G(x)`:

• Choice A)
  `F(x) = 3\;G(x+1)`;
• Choice B)
  `F(x) = 3\;G(x-1)`;
• Choice C)
  `F(x) = -3\;G(x+1)`;
• Choice D)
  `F(x) = -3\;G(x-1)`.

The expression in the braces (for example
`x` as in
`G(x)`) is the independent variable.

To shift a function on a cartesian plane to the left by
`a` units, add
`a` to its independent variable. Think about how
`(x-a)`, which is to the left of
`x`, will yield the same function value.

Conversely, to shift a function on a cartesian plane to the right by
`a` units, subtract
`a` from its independent variable.

For example,
`G(x+1)` is
`1` unit to the left of
`G(x)`. Conversely,
`G(x-1)` is
`1` unit to the right of
`G(x)`. The new function is to the left of
`G(x)`. Meaning that
`F(x)` should should add
`1` to (rather than subtract
`1` from) the independent variable of
`G(x)`. That rules out choice B) and D).

• Multiplying a function by a number that is greater than one will stretch its graph vertically.
• Multiplying a function by a number that is between zero and one will compress its graph vertically.
• Multiplying a function by a number that is between
  `-1` and zero will flip its graph about the
  `x`-axis. Doing so will also compress the graph vertically.
• Multiplying a function by a number that is less than
  `-1` will flip its graph about the
  `x`-axis. Doing so will also stretch the graph vertically.

The graph of
`G(x)` is stretched vertically. However, similarly to the graph of this graph
`G(x)`, the graph of
`F(x)` increases as
`x` increases. In other words, the graph of
`G(x)` isn't flipped about the
`x`-axis.
`G(x)` should have been multiplied by a number that is greater than one. That rules out choice C) and D).

Overall, only choice A) meets the requirements.

Since the plot in the question also came with a couple of gridlines, see if the points
`(x, y)`'s that are on the graph of
`F(x)` fit into the expression
`y = F(x) = 3√(x - 1)`.

Answer 2

f(x) =3 sqrt(x+1)

Explanation:

We notice two things about the graph, it has a shift to the left and is steeper

First the shift to the left

f(x) = g(x + C)

C > 0 moves it left

C < 0 moves it right

g(x) is 0 at x=0 f(x) is 0 at x=-1

We are moving it 1 unit to the left

This means our "c" is 1

f(x) = sqrt( x+1)

Now we need to deal with the graph getting steeper

f(x) = Cg(x)

C > 1 stretches it in the y-direction

0 < C < 1 compresses it

Since it is getting taller, "c" must be greater than 1

Remember the - sign means it is a reflection across the x axis, which we do not have

f(x) =3 sqrt(x+1)