Question

Please help with the attached question. Thanks

Answer 1

Answer:

Choice A) $F(x) = 3\sqrt{x + 1}$.

Step-by-step 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 $1$.)

$G(x) = \sqrt{x}$. 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 $y = F(x) = 3\sqrt{x - 1}$.

Answer 2

Answer:

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

Step-by-step 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)