The function  g(x)
  graphed in blue (solid) below is obtained by shifting the graph of  f(x)
   in red (dashed) vertically and horizontally.

 

Maple plot

 

 

The formula for  g(x)
  is

 

  g(x)=
  

To determine the formula for \( g(x) \) based on the given description that it's obtained by shifting \( f(x) \) both vertically and horizontally, we need to establish how the shift is applied.

The general form of shifting a function is:

\[
g(x) = f(x - h) + k
\]

Where:
- \( h \) represents the horizontal shift: if \( h > 0 \), the graph moves to the right; if \( h < 0 \), it moves to the left.
- \( k \) represents the vertical shift: if \( k > 0 \), the graph moves up; if \( k < 0 \), it moves down.

Thus, to complete the formula for \( g(x) \), we need the values of \( h \) and \( k \), which represent the horizontal and vertical shifts, respectively. These values can be determined by comparing the graph of \( g(x) \) (blue, solid) with that of \( f(x) \) (red, dashed).

If you provide more specific information about the graphs, such as a picture or additional description of the shifts, I can give you the exact formula.

Would you like more details or have any specific questions on this? 

Here are five questions to consider:
1. How is the graph of \( g(x) \) positioned relative to \( f(x) \)?
2. What are the coordinates of a key point on \( f(x) \) and the corresponding point on \( g(x) \)?
3. What does the function \( f(x) \) look like in its basic form?
4. How do the horizontal and vertical shifts affect the domain and range of the function?
5. Can we use symmetry or intercepts to better understand the shifts?

**Tip**: Horizontal shifts are often tricky because they move in the opposite direction of the sign. For example, \( f(x - 2) \) shifts the graph right, not left.

The function g(x) graphed in blue (solid) below is obtained by shifting the graph of f(x) in red (dashed) vertically and horizontally. The formula for g(x) is...

Explore how the function g(x) is obtained by shifting f(x) both horizontally and vertically. This problem covers key concepts such as horizontal and vertical shifts, with the formula g(x) = f(x - h) + k. Ideal for students in Grades 9-12.

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