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Изобрази схематический график функции

The exercise asks for the schematic graphs of the following functions:

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

b) \( y = \frac{1}{2}(x + 3)^2 - 1 \)

c) \( y = -4(x - 3)^2 + 5 \)

d) \( y = -4(x + 2)^2 - 2 \)

Each of these functions represents a transformation of the basic quadratic function \( y = x^2 \). Let’s analyze each one and its transformations:

### a) \( y = \frac{1}{2}(x - 2)^2 + 1 \)
- **Vertex**: The expression \( (x - 2) \) means the graph is shifted **2 units to the right**.
- **Vertical shift**: The \( +1 \) indicates a shift **1 unit up**.
- **Stretch**: The \( \frac{1}{2} \) indicates a **vertical compression** by a factor of \( \frac{1}{2} \).

The graph is a parabola opening upwards, centered at \( (2, 1) \), and is vertically compressed.

### b) \( y = \frac{1}{2}(x + 3)^2 - 1 \)
- **Vertex**: The \( (x + 3) \) means the graph is shifted **3 units to the left**.
- **Vertical shift**: The \( -1 \) shifts the graph **1 unit down**.
- **Stretch**: Similar to part a, the \( \frac{1}{2} \) causes a **vertical compression** by a factor of \( \frac{1}{2} \).

The graph is a parabola opening upwards, centered at \( (-3, -1) \), and is vertically compressed.

### c) \( y = -4(x - 3)^2 + 5 \)
- **Vertex**: The \( (x - 3) \) shifts the graph **3 units to the right**.
- **Vertical shift**: The \( +5 \) shifts the graph **5 units up**.
- **Stretch**: The coefficient \( -4 \) causes a **vertical stretch** by a factor of 4, and the negative sign makes the parabola open **downwards**.

The graph is a parabola opening downwards, centered at \( (3, 5) \), and is vertically stretched.

### d) \( y = -4(x + 2)^2 - 2 \)
- **Vertex**: The \( (x + 2) \) shifts the graph **2 units to the left**.
- **Vertical shift**: The \( -2 \) shifts the graph **2 units down**.
- **Stretch**: The coefficient \( -4 \) causes a **vertical stretch** and flips the parabola to open **downwards**.

The graph is a parabola opening downwards, centered at \( (-2, -2) \), and is vertically stretched.

These transformations help to sketch the graphs of the parabolas.

Do you need more details on how to graph these? Here are some follow-up questions:

1. How do we find the axis of symmetry for each quadratic function?
2. Can you explain the effect of a negative coefficient on the shape of the parabola?
3. What role does the vertex play in graphing a parabola?
4. How would these graphs change if the coefficients were positive instead of negative in cases c and d?
5. What is the significance of the vertical stretch/compression in each function?

**Tip**: Always plot the vertex first when sketching a parabola, and then use the stretch/compression factor to determine how "wide" or "narrow" the parabola will be.

Изобразите схематически график функции: a) y = (1/2)(x - 2)^2 + 1; б) y = (1/2)(x + 3)^2 - 1; в) y = -4(x - 3)^2 + 5; г) y = -4(x + 2)^2 - 2.

Learn how to graph quadratic functions of the form y = a(x - h)^2 + k, focusing on horizontal and vertical shifts, as well as vertical stretches and compressions. This problem explores the transformation of quadratic functions, including shifting and scaling.

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