To determine how the graph of function $f(x) = x - 3 + 1$ transforms into the graph of function $g(x) = x + 4 + 3$, we can analyze how the equations differ.

First, let's simplify both functions:

• $f(x) = x - 3 + 1 = x - 2$
• $g(x) = x + 4 + 3 = x + 7$

Now, we compare the two functions:

• $f(x) = x - 2$
• $g(x) = x + 7$

The transformation from $f(x)$ to $g(x)$ can be described by the changes in the constants:

• $f(x)$ has a constant of $-2$.
• $g(x)$ has a constant of $+7$.

The difference between the constants is: $7 - (-2) = 7 + 2 = 9$ So, the graph shifts 9 units up.

Now let’s break down the possible transformations based on the options:

• The transformation is a vertical shift, and it’s clear that the graph of $g(x)$ is shifted 9 units up from $f(x)$.

Thus, the correct statement is: The graph shifts 2 units left and 7 units up.

Further questions you might explore:

1. How do vertical and horizontal shifts differ in a graph transformation?
2. What happens if the constant term in the function is negative or positive?
3. How do the transformations change if you have a quadratic function instead of a linear one?
4. Can transformations be applied to other functions like absolute value or exponential functions?
5. How do you apply a reflection over the x-axis or y-axis to a function?

Tip:

When analyzing transformations, always simplify the functions and focus on how constants affect vertical shifts and coefficients affect horizontal shifts.