Final answer:
The function g(x) = (x + 4)^2 + 3 represents a horizontal shift to the left by 4 units and a vertical shift up by 3 units from the parent function f(x) = x^2, resulting in a new vertex at (-4, 3). There is no scaling, reflection, or rotation in this transformation.
Step-by-step explanation:
Description of the Transformation
The transformation g(x) = (x + 4)^2 + 3 involves several steps affecting the parent function f(x) = x^2, which is a parabola. The parent function is the simplest form of the parabola, and any transformation of this function will result in a shift, stretch, compression, or reflection of the original parabola.
Horizontal Shift
Firstly, the term (x + 4) inside the parentheses represents a horizontal shift. Normally, the vertex of the parent function f(x) = x^2 is at the origin (0, 0). However, adding 4 inside the square function translates the parabola to the left by 4 units, moving the vertex to the point (-4, 0).
Vertical Shift
Secondly, the + 3 outside the parentheses causes a vertical shift. After shifting the parabola horizontally, we then move it up by 3 units along the y-axis. This places the new vertex at (-4, 3), which is confirmed by completing the square in the equation of g(x).
No Scaling or Reflection
Because the coefficient of the squared term (x + 4)^2 is 1, there is no vertical stretch or compression, and the parabola remains as wide as the parent function. Additionally, as there is no negative sign in front of the squared term, there is no reflection over the x-axis, meaning that the parabola opens upwards just like the original f(x) = x^2.
Graphical Interpretation
Graphically, we can see these transformations by plotting the new function g(x) and comparing it to f(x). We will notice that every point on the graph of f(x) has been moved left by 4 units and up by 3 units to create the graph of g(x).
In summary, the transformation g(x) = (x + 4)^2 + 3 is a horizontal shift to the left by 4 units followed by a vertical shift upwards by 3 units from the parent function f(x) = x^2. There is no scaling, reflection, or rotation involved in this transformation.