To graph the function $g(x) = f(x+7)$, we need to understand how the transformation $f(x + 7)$ affects the graph of $f(x) = x$.

Step-by-Step Explanation:

1. Original Function $f(x) = x$:

   • The graph of $f(x) = x$ is a straight line passing through the origin with a slope of 1.
   • Its equation tells us that for each point $(x, f(x))$, the corresponding output is the same as the input, so the line rises at a 45-degree angle.

2. Transformation $g(x) = f(x+7)$:

   • This represents a horizontal shift.
   • The graph of $f(x + 7)$ shifts the original graph of $f(x) = x$ to the left by 7 units.
   • Every point on the graph of $f(x)$ is now shifted 7 units leftward.

Graphing:

• The line $f(x) = x$ has points like $(0, 0)$, $(1, 1)$, and $(2, 2)$.
• After the shift, the new graph $g(x) = f(x + 7)$ will have points such as:
  • $(-7, 0)$, $(-6, 1)$, and $(-5, 2)$.
• The slope remains the same (1), but the entire graph is moved to the left by 7 units.

Would you like me to create a visual representation of this graph?

Additional Questions:

1. How would $g(x) = f(x - 7)$ shift the graph?
2. What happens if we add a vertical shift, say $g(x) = f(x+7) + 3$?
3. How would the transformation affect the slope of the function if we had $f(x) = 2x$?
4. How do we represent a vertical reflection in such a graph?
5. How does a combination of horizontal and vertical shifts change the graph?

Tip:

Horizontal shifts occur when the transformation is applied inside the function (with $x$), while vertical shifts occur outside the function.