Identify the vertex and axis of symmetry for each quadratic function and sketch the graph.

Let’s solve each problem step by step. We’ll find the vertex and axis of symmetry, then describe how to sketch the graph.

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Problem 15:

f(x) = -3(x - 2)² - 4

This is in vertex form:
> f(x) = a(x - h)² + k
where (h, k) is the vertex.

Here:
- a = -3 → parabola opens downward
- h = 2
- k = -4

✔ Vertex: (2, -4)
✔ Axis of symmetry: x = 2 (vertical line through the vertex)

To sketch:
- Plot vertex at (2, -4)
- Since |a| = 3, it’s narrower than y = x²
- Opens downward
- Pick one point on either side of x=2 to help draw curve:
- Let x = 1 → f(1) = -3(1-2)² - 4 = -3(1) - 4 = -7 → point (1, -7)
- Let x = 3 → same as x=1 due to symmetry → (3, -7)
- Draw smooth U-shape opening down through these points.

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Problem 16:

f(x) = -¼(x - 1)² + 4

Vertex form again:
- a = -¼ → opens downward, very wide
- h = 1
- k = 4

✔ Vertex: (1, 4)
✔ Axis of symmetry: x = 1

To sketch:
- Plot vertex at (1, 4)
- Wide parabola opening down
- Try x = 0 → f(0) = -¼(-1)² + 4 = -¼ + 4 = 3.75 → (0, 3.75)
- x = 2 → same as x=0 → (2, 3.75)
- Maybe try x = -1 → f(-1) = -¼(-2)² + 4 = -¼(4) + 4 = -1 + 4 = 3 → (-1, 3)
- Symmetric point: x = 3 → also 3 → (3, 3)
- Connect with smooth wide downward curve.

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Problem 17:

f(x) = ¼(x + 4)² + 3

Rewrite as: f(x) = ¼(x - (-4))² + 3

So:
- a = ¼ → opens upward, wide
- h = -4
- k = 3

✔ Vertex: (-4, 3)
✔ Axis of symmetry: x = -4

To sketch:
- Plot vertex at (-4, 3)
- Try x = -2 → f(-2) = ¼(-2+4)² + 3 = ¼(4) + 3 = 1 + 3 = 4 → (-2, 4)
- x = -6 → same → (-6, 4)
- x = 0 → f(0) = ¼(4)² + 3 = ¼(16) + 3 = 4 + 3 = 7 → (0, 7)
- x = -8 → same → (-8, 7)
- Draw wide upward U-shape.

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Problem 18:

f(x) = ¼(x + 5)² + 2

Rewrite: f(x) = ¼(x - (-5))² + 2

So:
- a = ¼ → opens upward, wide
- h = -5
- k = 2

✔ Vertex: (-5, 2)
✔ Axis of symmetry: x = -5

To sketch:
- Plot vertex at (-5, 2)
- Try x = -3 → f(-3) = ¼(-3+5)² + 2 = ¼(4) + 2 = 1 + 2 = 3 → (-3, 3)
- x = -7 → same → (-7, 3)
- x = -1 → f(-1) = ¼(-1+5)² + 2 = ¼(16) + 2 = 4 + 2 = 6 → (-1, 6)
- x = -9 → same → (-9, 6)
- Draw wide upward U-shape.

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## Final Answer:

15)
Vertex: (2, -4)
Axis of symmetry: x = 2

16)
Vertex: (1, 4)
Axis of symmetry: x = 1

17)
Vertex: (-4, 3)
Axis of symmetry: x = -4

18)
Vertex: (-5, 2)
Axis of symmetry: x = -5

*(Graphs should be sketched using the vertex, axis of symmetry, and additional calculated points as shown above.)*