Answer :
To determine the graph of the function [tex]\( g(x) = (x-2)^2 \)[/tex] using transformations of the graph of [tex]\( f(x) = x^2 \)[/tex], we will follow a step-by-step process to understand how the graph of [tex]\( f(x) \)[/tex] changes to become the graph of [tex]\( g(x) \)[/tex].
### Step-by-Step Transformation:
1. Identify the Base Function:
- The base function is [tex]\( f(x) = x^2 \)[/tex].
- This is a standard parabola that opens upwards and has its vertex at the origin [tex]\((0, 0)\)[/tex].
2. Understand the Transformation:
- The transformation given is [tex]\( g(x) = (x-2)^2 \)[/tex].
- This specific form indicates that the transformation involves a horizontal shift.
3. Determine the Nature of the Shift:
- In the function [tex]\( g(x) = (x - 2)^2 \)[/tex], the [tex]\((x - 2)\)[/tex] inside the function causes a horizontal shift to the right.
- The general form [tex]\( (x - h) \)[/tex] represents a horizontal shift by [tex]\( h \)[/tex] units to the right if [tex]\( h \)[/tex] is positive and to the left if [tex]\( h \)[/tex] is negative.
4. Apply the Horizontal Shift:
- Here, [tex]\( h = 2 \)[/tex] because we have [tex]\( (x - 2) \)[/tex].
- Therefore, every point on the graph of [tex]\( f(x) = x^2 \)[/tex] will be shifted 2 units to the right.
### Constructing the New Graph:
- The vertex of [tex]\( f(x) = x^2 \)[/tex] is at [tex]\((0, 0)\)[/tex].
- Applying the horizontal shift of 2 units to the right:
- The new vertex of [tex]\( g(x) = (x-2)^2 \)[/tex] will be at [tex]\((2, 0)\)[/tex].
- For other points on the graph:
- Consider a point [tex]\( (a, f(a)) \)[/tex] on the graph of [tex]\( f(x) \)[/tex].
- After shifting this point 2 units to the right, the new point becomes [tex]\( (a+2, f(a)) \)[/tex] for the graph of [tex]\( g(x) \)[/tex].
- Example of points for illustration:
- Point [tex]\((1, 1)\)[/tex] on [tex]\( f(x) \)[/tex] becomes [tex]\((3, 1)\)[/tex] on [tex]\( g(x) \)[/tex].
- Point [tex]\((-1, 1)\)[/tex] on [tex]\( f(x) \)[/tex] becomes [tex]\((1, 1)\)[/tex] on [tex]\( g(x) \)[/tex].
### Summary of Transformation:
- The original function [tex]\( f(x) = x^2 \)[/tex] represents a parabola centered at [tex]\((0, 0)\)[/tex].
- The function [tex]\( g(x) = (x-2)^2 \)[/tex] represents the same parabola shifted 2 units to the right, centering the vertex at [tex]\((2, 0)\)[/tex].
Thus, the graph of [tex]\( g(x) = (x-2)^2 \)[/tex] is the graph of [tex]\( f(x) = x^2 \)[/tex] shifted 2 units to the right.
### Step-by-Step Transformation:
1. Identify the Base Function:
- The base function is [tex]\( f(x) = x^2 \)[/tex].
- This is a standard parabola that opens upwards and has its vertex at the origin [tex]\((0, 0)\)[/tex].
2. Understand the Transformation:
- The transformation given is [tex]\( g(x) = (x-2)^2 \)[/tex].
- This specific form indicates that the transformation involves a horizontal shift.
3. Determine the Nature of the Shift:
- In the function [tex]\( g(x) = (x - 2)^2 \)[/tex], the [tex]\((x - 2)\)[/tex] inside the function causes a horizontal shift to the right.
- The general form [tex]\( (x - h) \)[/tex] represents a horizontal shift by [tex]\( h \)[/tex] units to the right if [tex]\( h \)[/tex] is positive and to the left if [tex]\( h \)[/tex] is negative.
4. Apply the Horizontal Shift:
- Here, [tex]\( h = 2 \)[/tex] because we have [tex]\( (x - 2) \)[/tex].
- Therefore, every point on the graph of [tex]\( f(x) = x^2 \)[/tex] will be shifted 2 units to the right.
### Constructing the New Graph:
- The vertex of [tex]\( f(x) = x^2 \)[/tex] is at [tex]\((0, 0)\)[/tex].
- Applying the horizontal shift of 2 units to the right:
- The new vertex of [tex]\( g(x) = (x-2)^2 \)[/tex] will be at [tex]\((2, 0)\)[/tex].
- For other points on the graph:
- Consider a point [tex]\( (a, f(a)) \)[/tex] on the graph of [tex]\( f(x) \)[/tex].
- After shifting this point 2 units to the right, the new point becomes [tex]\( (a+2, f(a)) \)[/tex] for the graph of [tex]\( g(x) \)[/tex].
- Example of points for illustration:
- Point [tex]\((1, 1)\)[/tex] on [tex]\( f(x) \)[/tex] becomes [tex]\((3, 1)\)[/tex] on [tex]\( g(x) \)[/tex].
- Point [tex]\((-1, 1)\)[/tex] on [tex]\( f(x) \)[/tex] becomes [tex]\((1, 1)\)[/tex] on [tex]\( g(x) \)[/tex].
### Summary of Transformation:
- The original function [tex]\( f(x) = x^2 \)[/tex] represents a parabola centered at [tex]\((0, 0)\)[/tex].
- The function [tex]\( g(x) = (x-2)^2 \)[/tex] represents the same parabola shifted 2 units to the right, centering the vertex at [tex]\((2, 0)\)[/tex].
Thus, the graph of [tex]\( g(x) = (x-2)^2 \)[/tex] is the graph of [tex]\( f(x) = x^2 \)[/tex] shifted 2 units to the right.
