https://file.aiquickdraw.com/mathbot/file/1727052364984oo1gh6z6.jpg

State transformations, sketch graph, state domain and range and other points

The equation given in the image is:

\[
y = \frac{1}{2}(x - 2)^2 + 1
\]

This represents a **parabola** in vertex form. Let's break down the components, transformations, domain and range, and sketch the graph.

### 1. **Transformations:**
   The equation \( y = a(x - h)^2 + k \) is the standard vertex form of a parabola.
   
   - **Vertex**: The vertex is \((h, k)\). In this case, the vertex is at \((2, 1)\).
   - **Vertical Stretch/Compression**: The coefficient \( a = \frac{1}{2} \) indicates a vertical compression by a factor of \( \frac{1}{2} \) (making the parabola wider than the standard parabola \( y = x^2 \)).
   - **Horizontal Shift**: The term \( (x - 2) \) indicates a horizontal shift 2 units to the right.
   - **Vertical Shift**: The \( +1 \) means the graph is shifted 1 unit up.

### 2. **Sketch of the Graph:**
   The parabola opens **upwards** because the leading coefficient \( \frac{1}{2} \) is positive. Here's a general description:
   - The vertex is at \( (2, 1) \), which is the minimum point.
   - The parabola is wider than the standard one due to the vertical compression.

### 3. **Domain and Range:**
   - **Domain**: Since this is a quadratic equation, the domain is all real numbers: \( x \in (-\infty, \infty) \).
   - **Range**: Since the parabola opens upwards and has a minimum at \( y = 1 \), the range is \( y \in [1, \infty) \).

### 4. **Other Key Points:**
   - At \( x = 2 \), \( y = 1 \) (the vertex).
   - When \( x = 0 \), plug into the equation:
     \[
     y = \frac{1}{2}(0 - 2)^2 + 1 = \frac{1}{2}(4) + 1 = 2 + 1 = 3
     \]
     So, another point is \( (0, 3) \).
   - Similarly, at \( x = 4 \):
     \[
     y = \frac{1}{2}(4 - 2)^2 + 1 = \frac{1}{2}(2)^2 + 1 = \frac{1}{2}(4) + 1 = 2 + 1 = 3
     \]
     So, \( (4, 3) \) is also a point on the parabola.

Would you like to go over any specific part of the graph or calculation?

### Related Questions:
1. What effect would increasing the coefficient from \( \frac{1}{2} \) to 2 have on the parabola?
2. How do we determine if a parabola opens upwards or downwards based on the equation?
3. Can you find the equation of a parabola with vertex at \( (-3, -2) \) and opening downwards?
4. What is the general approach to finding the axis of symmetry of a parabola?
5. How do we calculate the x-intercepts of the parabola if it has any?

### Tip:
When graphing quadratics, always start by identifying the vertex and direction of the parabola based on the leading coefficient. This will help set the foundation for the graph's shape.

State transformations, sketch graph, state domain and range and other points.

Explore how to transform the quadratic function y = (1/2)(x - 2)^2 + 1. This problem covers key algebraic concepts such as horizontal and vertical shifts, vertical compression, and how to determine the domain and range of a parabola. Suitable for students in Grades 9-11.

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