An answer key displaying nine graphed quadratic functions with their corresponding vertices and axes of symmetry identified in red.

It looks like you've uploaded an image of a worksheet titled "Graphing Quadratic Functions: f(x) = ax² + bx + c – ANSWER KEY". This is a completed answer key for a set of quadratic function graphing problems, where students are asked to identify the vertex and axis of symmetry for each given quadratic equation.

Let me walk through how to solve these types of problems and explain the logic behind the answers shown in the key.

---

🔷 General Form of a Quadratic Function:

$$
f(x) = ax^2 + bx + c
$$

For any quadratic function in this form:

- The axis of symmetry is given by:
$$
x = -\frac{b}{2a}
$$

- The vertex lies on the axis of symmetry. To find the y-coordinate of the vertex, plug the x-value back into the function:
$$
f\left(-\frac{b}{2a}\right)
$$

So, the vertex is:
$$
\left(-\frac{b}{2a},\ f\left(-\frac{b}{2a}\right)\right)
$$

---

Now let's go through a few examples from your worksheet and verify the answers.

---

✔ Example 1:

Function: $ f(x) = -x^2 + 4x - 3 $

- $ a = -1, b = 4 $
- Axis of symmetry:
$$
x = -\frac{4}{2(-1)} = -\frac{4}{-2} = 2
$$
- Vertex: Plug $ x = 2 $ into $ f(x) $:
$$
f(2) = -(2)^2 + 4(2) - 3 = -4 + 8 - 3 = 1
$$
So, vertex = $ (2, 1) $

✔ Matches the key: Vertex: (2, 1), Axis of Symmetry: x = 2

---

✔ Example 2:

Function: $ f(x) = x^2 + 6x + 5 $

- $ a = 1, b = 6 $
- Axis of symmetry:
$$
x = -\frac{6}{2(1)} = -3
$$
- $ f(-3) = (-3)^2 + 6(-3) + 5 = 9 - 18 + 5 = -4 $
- Vertex = $ (-3, -4) $

✔ Matches the key: Vertex: (-3, -4), Axis of Symmetry: x = -3

---

✔ Example 3:

Function: $ f(x) = x^2 - 4x + 12 $

- $ a = 1, b = -4 $
- Axis of symmetry:
$$
x = -\frac{-4}{2(1)} = \frac{4}{2} = 2
$$
- $ f(2) = (2)^2 - 4(2) + 12 = 4 - 8 + 12 = 8 $
- Vertex = $ (2, 8) $

✔ Matches the key: Vertex: (2, 8), Axis of Symmetry: x = 2

---

✔ Example 4:

Function: $ f(x) = -2x^2 + 4x + 7 $

- $ a = -2, b = 4 $
- Axis of symmetry:
$$
x = -\frac{4}{2(-2)} = -\frac{4}{-4} = 1
$$
- $ f(1) = -2(1)^2 + 4(1) + 7 = -2 + 4 + 7 = 9 $
- Vertex = $ (1, 9) $

✔ Matches the key: Vertex: (1, 9), Axis of Symmetry: x = 1

---

✔ Example 5:

Function: $ f(x) = -2x^2 - 8x - 30 $

- $ a = -2, b = -8 $
- Axis of symmetry:
$$
x = -\frac{-8}{2(-2)} = -\frac{-8}{-4} = -2
$$
- $ f(-2) = -2(-2)^2 - 8(-2) - 30 = -2(4) + 16 - 30 = -8 + 16 - 30 = -22 $
- Vertex = $ (-2, -22) $

✔ Matches the key: Vertex: (-2, -22), Axis of Symmetry: x = -2

---

✔ Example 6:

Function: $ f(x) = x^2 + 4x + 8 $

- $ a = 1, b = 4 $
- Axis of symmetry:
$$
x = -\frac{4}{2(1)} = -2
$$
- $ f(-2) = (-2)^2 + 4(-2) + 8 = 4 - 8 + 8 = 4 $
- Vertex = $ (-2, 4) $

✔ Matches the key: Vertex: (-2, 4), Axis of Symmetry: x = -2

---

✔ Example 7:

Function: $ f(x) = \frac{1}{2}x^2 + x + 8 $

- $ a = \frac{1}{2}, b = 1 $
- Axis of symmetry:
$$
x = -\frac{1}{2(\frac{1}{2})} = -\frac{1}{1} = -1
$$
- $ f(-1) = \frac{1}{2}(1) + (-1) + 8 = 0.5 - 1 + 8 = 7.5 $
- Vertex = $ (-1, 7.5) $ or $ (-1, \frac{15}{2}) $

✔ Matches the key: Vertex: (-1, 7.5), Axis of Symmetry: x = -1

---

✔ Example 8:

Function: $ f(x) = 4x^2 + 8x + 20 $

- $ a = 4, b = 8 $
- Axis of symmetry:
$$
x = -\frac{8}{2(4)} = -\frac{8}{8} = -1
$$
- $ f(-1) = 4(1) + 8(-1) + 20 = 4 - 8 + 20 = 16 $
- Vertex = $ (-1, 16) $

✔ Matches the key: Vertex: (-1, 16), Axis of Symmetry: x = -1

---

✔ Example 9:

Function: $ f(x) = -\frac{1}{2}x^2 - 2x + 4 $

- $ a = -\frac{1}{2}, b = -2 $
- Axis of symmetry:
$$
x = -\frac{-2}{2(-\frac{1}{2})} = -\frac{-2}{-1} = -2
$$
- $ f(-2) = -\frac{1}{2}(4) - 2(-2) + 4 = -2 + 4 + 4 = 6 $
- Vertex = $ (-2, 6) $

✔ Matches the key: Vertex: (-2, 6), Axis of Symmetry: x = -2

---

## 📌 Summary of Steps to Solve These Problems:

1. Identify $ a $, $ b $, and $ c $ in $ f(x) = ax^2 + bx + c $.
2. Use $ x = -\frac{b}{2a} $ to find the axis of symmetry.
3. Plug that $ x $-value back into the function to get the y-coordinate of the vertex.
4. Write the vertex as $ (x, y) $ and the axis of symmetry as $ x = \text{value} $.

---

All the answers in the provided key match the correct calculations. The graphs also reflect the correct shape and orientation based on the sign of $ a $:
- If $ a > 0 $ → parabola opens up
- If $ a < 0 $ → parabola opens down

The vertex is the minimum (if up) or maximum (if down) point.

---

✔ Final Note: The answer key appears to be correct and well-formatted. Each problem follows the standard method for finding vertex and axis of symmetry.

Let me know if you'd like help creating a similar worksheet or solving a new quadratic!