The given function is $g(x) = (x + 4)^2$, and you are asked to use transformations of the graph of $f(x) = x^2$ to determine the graph of $g(x)$.

Step-by-Step Transformation:

1. Starting with the basic function:

   • The base function is $f(x) = x^2$, which is a standard parabola with its vertex at the origin $(0, 0)$ and opens upwards.

2. Horizontal shift:

   • The function $g(x) = (x + 4)^2$ involves a horizontal shift. When we add a value inside the parentheses like this ($x + 4$), it causes a horizontal shift to the left by 4 units.
   • This means the vertex of the parabola, which was originally at $(0, 0)$, will now be at $(-4, 0)$.

3. Shape and orientation:

   • The shape of the graph of $g(x)$ remains the same as $f(x) = x^2$. It is still a parabola opening upwards since the coefficient of $(x + 4)^2$ is positive.
   • There is no vertical stretch, compression, or reflection involved here, as the equation remains $(x + 4)^2$.

Summary:

• The graph of $g(x) = (x + 4)^2$ is a parabola that has been shifted 4 units to the left from the graph of $f(x) = x^2$. The vertex of the new parabola is at $(-4, 0)$.

Would you like further details on this transformation or have any questions?

Here are some follow-up questions that might help expand your understanding:

1. How would the graph change if the equation was $g(x) = (x - 4)^2$?
2. What would happen if the equation included a vertical shift, like $g(x) = (x + 4)^2 + 2$?
3. How does a negative coefficient, such as $g(x) = -(x + 4)^2$, affect the graph?
4. How can you determine the axis of symmetry of the function $g(x) = (x + 4)^2$?
5. What is the effect of adding a coefficient in front of the squared term, such as $g(x) = 2(x + 4)^2$?

Tip: Horizontal shifts always move the graph in the opposite direction of the sign inside the function. For example, $x + 4$ shifts left, and $x - 4$ shifts right.