Axis of symmetry worksheet | TPT - Free Printable

Since I can't view or access uploaded images directly, I can’t see the specific graphs or equations in your image. However, based on the text you've provided, I can help you solve the problems described in Part 1, Part 2, and Part 3 about finding the axis of symmetry for quadratic functions and their graphs.

Let me walk through each part step by step and explain how to solve them.

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🔷 Part 1: Determine the axis of symmetry from the graph

The axis of symmetry is a vertical line that divides a parabola into two mirror-image halves. For a parabola given by $ y = ax^2 + bx + c $, the axis of symmetry is:

$$
x = -\frac{b}{2a}
$$

But since this part asks you to determine it from the graph, you need to:
- Look at the vertex (the highest or lowest point).
- The axis of symmetry passes through the vertex’s x-coordinate.
- So, if the vertex is at $ (h, k) $, then the axis of symmetry is $ x = h $.

Without seeing the graphs, I can't give exact answers, but here's how you'd do it:

#### Example:
If a graph has its vertex at $ (2, 4) $, then the axis of symmetry is $ x = 2 $.

So for each graph labeled A–E:
- Identify the vertex.
- Write down the x-coordinate as the axis of symmetry.

> ⚠️ Since I can't see the graphs, you’ll need to do this visually.

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🔷 Part 2: Calculate the axis of symmetry given the equation

For each quadratic function $ f(x) = ax^2 + bx + c $, use the formula:

$$
x = -\frac{b}{2a}
$$

Let’s go through each one:

#### 1) $ f(x) = x^2 + 4x + 3 $
- $ a = 1, b = 4 $
- $ x = -\frac{4}{2(1)} = -2 $
✔ Axis of symmetry: $ x = -2 $

#### 2) $ f(x) = x^2 - 6 $
- $ a = 1, b = 0 $
- $ x = -\frac{0}{2(1)} = 0 $
✔ Axis of symmetry: $ x = 0 $

#### 3) $ f(x) = x^2 + 5x - 3 $
- $ a = 1, b = 5 $
- $ x = -\frac{5}{2(1)} = -2.5 $
✔ Axis of symmetry: $ x = -2.5 $

#### 4) $ f(x) = 2x^2 - 8x + 3 $
- $ a = 2, b = -8 $
- $ x = -\frac{-8}{2(2)} = \frac{8}{4} = 2 $
✔ Axis of symmetry: $ x = 2 $

#### 5) $ f(x) = -3x^2 + 12x - 6 $
- $ a = -3, b = 12 $
- $ x = -\frac{12}{2(-3)} = -\frac{12}{-6} = 2 $
✔ Axis of symmetry: $ x = 2 $

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

#### 7) $ f(x) = x^2 + 2x - 6 $
- $ a = 1, b = 2 $
- $ x = -\frac{2}{2(1)} = -1 $
✔ Axis of symmetry: $ x = -1 $

#### 8) $ f(x) = -x^2 + 6x + 9 $
- $ a = -1, b = 6 $
- $ x = -\frac{6}{2(-1)} = -\frac{6}{-2} = 3 $
✔ Axis of symmetry: $ x = 3 $

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🔷 Part 3: Find the axis of symmetry by completing the square

We will rewrite each quadratic in vertex form:
$$
f(x) = a(x - h)^2 + k
$$
Then the axis of symmetry is $ x = h $

Let’s go through each:

#### 1) $ f(x) = x^2 + 4x + 3 $
- $ x^2 + 4x = (x+2)^2 - 4 $
- So: $ f(x) = (x+2)^2 - 4 + 3 = (x+2)^2 - 1 $
- Vertex: $ (-2, -1) $
✔ Axis: $ x = -2 $

#### 2) $ f(x) = x^2 + 10x + 15 $
- $ x^2 + 10x = (x+5)^2 - 25 $
- $ f(x) = (x+5)^2 - 25 + 15 = (x+5)^2 - 10 $
✔ Axis: $ x = -5 $

#### 3) $ f(x) = x^2 + 10x - 15 $
- $ x^2 + 10x = (x+5)^2 - 25 $
- $ f(x) = (x+5)^2 - 25 - 15 = (x+5)^2 - 40 $
✔ Axis: $ x = -5 $

#### 4) $ f(x) = x^2 - 3x + 24 $
- $ x^2 - 3x = (x - 1.5)^2 - 2.25 $
- $ f(x) = (x - 1.5)^2 - 2.25 + 24 = (x - 1.5)^2 + 21.75 $
✔ Axis: $ x = 1.5 $

#### 5) $ f(x) = x^2 - 5 $
- Already almost in vertex form: $ f(x) = (x - 0)^2 - 5 $
✔ Axis: $ x = 0 $

#### 6) $ f(x) = x^2 + 6x + 8 $
- $ x^2 + 6x = (x+3)^2 - 9 $
- $ f(x) = (x+3)^2 - 9 + 8 = (x+3)^2 - 1 $
✔ Axis: $ x = -3 $

#### 7) $ f(x) = x^2 + 2x + 3 $
- $ x^2 + 2x = (x+1)^2 - 1 $
- $ f(x) = (x+1)^2 - 1 + 3 = (x+1)^2 + 2 $
✔ Axis: $ x = -1 $

#### 8) $ f(x) = x^2 - 8x + 16 $
- $ x^2 - 8x = (x - 4)^2 - 16 $
- $ f(x) = (x - 4)^2 - 16 + 16 = (x - 4)^2 $
✔ Axis: $ x = 4 $

#### 9) $ f(x) = 2x^2 + 4x - 3 $
- Factor out 2: $ 2(x^2 + 2x) - 3 $
- $ x^2 + 2x = (x+1)^2 - 1 $
- So: $ 2[(x+1)^2 - 1] - 3 = 2(x+1)^2 - 2 - 3 = 2(x+1)^2 - 5 $
✔ Axis: $ x = -1 $

#### 10) $ f(x) = 3x^2 + 6x + 1 $
- Factor out 3: $ 3(x^2 + 2x) + 1 $
- $ x^2 + 2x = (x+1)^2 - 1 $
- $ 3[(x+1)^2 - 1] + 1 = 3(x+1)^2 - 3 + 1 = 3(x+1)^2 - 2 $
✔ Axis: $ x = -1 $

#### 11) $ f(x) = x^2 + 2x + 2 $
- $ x^2 + 2x = (x+1)^2 - 1 $
- $ f(x) = (x+1)^2 - 1 + 2 = (x+1)^2 + 1 $
✔ Axis: $ x = -1 $

#### 12) $ f(x) = x^2 + 12x + 21 $
- $ x^2 + 12x = (x+6)^2 - 36 $
- $ f(x) = (x+6)^2 - 36 + 21 = (x+6)^2 - 15 $
✔ Axis: $ x = -6 $

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✔ Summary of Answers

#### Part 2:
| Function | Axis of Symmetry |
|--------|------------------|
| 1) $ x^2 + 4x + 3 $ | $ x = -2 $ |
| 2) $ x^2 - 6 $ | $ x = 0 $ |
| 3) $ x^2 + 5x - 3 $ | $ x = -2.5 $ |
| 4) $ 2x^2 - 8x + 3 $ | $ x = 2 $ |
| 5) $ -3x^2 + 12x - 6 $ | $ x = 2 $ |
| 6) $ x^2 - 4x + 4 $ | $ x = 2 $ |
| 7) $ x^2 + 2x - 6 $ | $ x = -1 $ |
| 8) $ -x^2 + 6x + 9 $ | $ x = 3 $ |

#### Part 3:
| Function | Axis of Symmetry |
|--------|------------------|
| 1) $ x^2 + 4x + 3 $ | $ x = -2 $ |
| 2) $ x^2 + 10x + 15 $ | $ x = -5 $ |
| 3) $ x^2 + 10x - 15 $ | $ x = -5 $ |
| 4) $ x^2 - 3x + 24 $ | $ x = 1.5 $ |
| 5) $ x^2 - 5 $ | $ x = 0 $ |
| 6) $ x^2 + 6x + 8 $ | $ x = -3 $ |
| 7) $ x^2 + 2x + 3 $ | $ x = -1 $ |
| 8) $ x^2 - 8x + 16 $ | $ x = 4 $ |
| 9) $ 2x^2 + 4x - 3 $ | $ x = -1 $ |
| 10) $ 3x^2 + 6x + 1 $ | $ x = -1 $ |
| 11) $ x^2 + 2x + 2 $ | $ x = -1 $ |
| 12) $ x^2 + 12x + 21 $ | $ x = -6 $ |

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📌 Final Notes:

- The axis of symmetry is always $ x = -\frac{b}{2a} $ for any quadratic.
- Completing the square gives the same result and helps visualize the vertex.
- For Part 1, you must estimate the vertex from the graph and report the x-value.

If you can describe the graphs in Part 1 (e.g., “Graph A has vertex at (3, 2)” etc.), I can help identify those too.

Let me know if you want these answers formatted neatly or explained further!