https://file.aiquickdraw.com/mathbot/file/1723385882202wk4kypj4.jpg

The image contains six quadratic functions that can be analyzed to determine their key characteristics, such as vertex, axis of symmetry, and the type of parabola (opening upwards or downwards). Below are the equations:

1. \( f(x) = (x + 4)^2 - 3 \)
2. \( f(x) = 5(x - 1)^2 - 20 \)
3. \( f(x) = 3x^2 + 7 \)
4. \( f(x) = x^2 + 12x + 11 \)
5. \( f(x) = 2x^2 - 12x + 26 \)
6. \( f(x) = 4x^2 - 4x + 1 \)

Let's analyze each one:

### 1. \( f(x) = (x + 4)^2 - 3 \)
- **Vertex Form**: The equation is already in vertex form, \( f(x) = a(x-h)^2 + k \), where the vertex \((h, k)\) is \((-4, -3)\).
- **Axis of Symmetry**: \( x = -4 \)
- **Direction**: Since \(a = 1\), the parabola opens upwards.

### 2. \( f(x) = 5(x - 1)^2 - 20 \)
- **Vertex Form**: The vertex is \((1, -20)\).
- **Axis of Symmetry**: \( x = 1 \)
- **Direction**: Since \(a = 5\), the parabola opens upwards and is narrower than the standard parabola.

### 3. \( f(x) = 3x^2 + 7 \)
- **Standard Form**: This can be seen as \( f(x) = 3(x - 0)^2 + 7 \), so the vertex is \((0, 7)\).
- **Axis of Symmetry**: \( x = 0 \)
- **Direction**: Since \(a = 3\), the parabola opens upwards and is narrower than the standard parabola.

### 4. \( f(x) = x^2 + 12x + 11 \)
- **Standard Form**: To find the vertex, complete the square or use \( x = -\frac{b}{2a} \):
  \[
  x = -\frac{12}{2(1)} = -6
  \]
  Plug \( x = -6 \) back into the function to find the y-coordinate:
  \[
  f(-6) = (-6)^2 + 12(-6) + 11 = 36 - 72 + 11 = -25
  \]
  Vertex: \((-6, -25)\)
- **Axis of Symmetry**: \( x = -6 \)
- **Direction**: Since \(a = 1\), the parabola opens upwards.

### 5. \( f(x) = 2x^2 - 12x + 26 \)
- **Standard Form**: Again, use \( x = -\frac{b}{2a} \):
  \[
  x = -\frac{-12}{2(2)} = \frac{12}{4} = 3
  \]
  Plug \( x = 3 \) back into the function:
  \[
  f(3) = 2(3)^2 - 12(3) + 26 = 18 - 36 + 26 = 8
  \]
  Vertex: \((3, 8)\)
- **Axis of Symmetry**: \( x = 3 \)
- **Direction**: Since \(a = 2\), the parabola opens upwards.

### 6. \( f(x) = 4x^2 - 4x + 1 \)
- **Standard Form**: Use \( x = -\frac{b}{2a} \):
  \[
  x = -\frac{-4}{2(4)} = \frac{4}{8} = 0.5
  \]
  Plug \( x = 0.5 \) back into the function:
  \[
  f(0.5) = 4(0.5)^2 - 4(0.5) + 1 = 1 - 2 + 1 = 0
  \]
  Vertex: \((0.5, 0)\)
- **Axis of Symmetry**: \( x = 0.5 \)
- **Direction**: Since \(a = 4\), the parabola opens upwards and is narrower.

Would you like to go into further detail on any of these points or explore related concepts?

Here are some related questions you might consider:
1. How do you complete the square to convert a quadratic equation from standard form to vertex form?
2. How do you determine the direction of a parabola from the coefficient of \(x^2\)?
3. What does the axis of symmetry represent in a quadratic function?
4. How does the value of the coefficient \(a\) affect the width of a parabola?
5. Can you explain the process to find the vertex using the vertex formula?
6. What happens to the parabola if the coefficient \(a\) is negative?
7. How do you find the y-intercept of a quadratic function?
8. Can you describe the significance of the vertex in the graph of a quadratic function?

**Tip:** When dealing with quadratic functions, the vertex form \(f(x) = a(x-h)^2 + k\) is particularly useful for quickly identifying the vertex and the direction in which the parabola opens.

Explore six quadratic functions and learn to analyze their vertex, axis of symmetry, and direction of parabolas. This detailed analysis covers equations like f(x) = (x + 4)^2 - 3 and f(x) = 2x^2 - 12x + 26. Suitable for high school students studying quadratic functions.

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