Question

"The graph of F(x), shown below, resembles the graph of G(x) = x2, but it has been changed somewhat. Which of the following could be the equation of F(x)?

Answer 1

G(x)=x²
The graph  has moved to the right 4 units, therefore the new graph will be:
H(x)=(x-4)²

It has also move 4 units up, therefore the new graph will be:
F(x)=(x-4)²+4

Answer: 
F(x)=(x-4)²+4

Answer 2

y = (x - 4)² + 4

or y = x² - 8x + 20

Further explanation

Transformation of a graph is changing the shape and location of a graph.

There are four types of transformation geometry: translation (or shifting), reflection, rotation, and dilation (or stretching).  

• In this case, the transformation is shifting horizontally or vertically.
• Translation (or shifting): moving a graph on an analytic plane without changing its shape.
• Vertical shift: moving a graph upwards or downwards without changing its shape.
• Horizontal shift: moving a graph to the left or right downwards without changing its shape.  

In general, given the graph of y = f(x) and v > 0, we obtain the graph of:

• $\boxed{ \ y = f(x) + v \ }$ by shifting the graph of $\boxed{ \ y = f(x) \ }$ upward v units.
• $\boxed{ \ y = f(x) - v \ }$ by shifting the graph of $\boxed{ \ y = f(x) \ }$ downward v units.

That's the vertical shift, now the horizontal one. Given the graph of y = f(x) and h > 0, we obtain the graph of:

• $\boxed{ \ y = f(x + h) \ }$ by shifting the graph of $\boxed{ \ y = f(x) \ }$ to the left h units.
• $\boxed{ \ y = f(x - h) \ }$ by shifting the graph of $\boxed{ \ y = f(x) \ }$ to the right h units.

Hence, the combination of vertical and horizontal shifts is as follows:

$\boxed{\boxed{ \ y = f(x \pm h) \pm v \ }}$

The plus or minus sign follows the direction of the shift, i.e., up-down or left-right

Given: $\boxed{ \ g(x) = x^2 \ becomes \ f(x) = ? \ }$

In the graph, notice the shifting of the vertex from (0, 0) to (4, 4).

From this, we can describe that from g(x) to f(x) there has been a shift to the right 4 units and upward 4 units.

Let us construct f(x) from g(x).

$\boxed{ \ g(x) = y = x^2 \ } \rightarrow \boxed{ \ f(x) = y = (x + h)^2 + v \ }$

We set h = -4 and v = +4 and we get the equation f(x) as

$\boxed{\boxed{ \ f(x) = (x - 4)^2 + 4 \ }}$

Let's expand it if we want to represent a standard form of a quadratic function, like this:

$\boxed{ \ f(x) = x^2 - 8x + 16 + 4 \ }$

$\boxed{\boxed{ \ f(x) = x^2 - 8x + 20 \ }}$

Conclusion

The graph of f(x) is drawn by the combination of shifting the graph of g(x) to the right 4 units and upward 4 units.  

Learn more  

1. Transformations that change the graph of (f)x to the graph of g(x) brainly.com/question/2415963
2. The similar problem brainly.com/question/1369568
3. Determine the coordinates of the image of a point after the triangle is rotated 270° about the origin brainly.com/question/7437053

Keywords: transformations, the graph of f(x), resembles, g(x) = x², f(x) = (x - 4)² + 4, y = x² - 8x + 20, translation, shifting, right, upward , horizontal, vertical