Graphs of Parabolas - Vertex Form | PDF - Free Printable

I'll solve each parabola problem step by step, identifying all the required characteristics.

## Problem 1: y = (x - 5)² - 4

Vertex: In vertex form y = a(x - h)² + k, the vertex is (h, k)
- h = 5, k = -4
- Vertex: (5, -4)

Axis of symmetry: x = h
- Axis of symmetry: x = 5

Direction of opening: Look at 'a' (coefficient in front)
- a = 1 (positive), so opens upward

Min/Max value: Since it opens upward, there's a minimum
- Minimum value: -4 (at the vertex)

Y-intercept: Set x = 0
- y = (0 - 5)² - 4 = 25 - 4 = 21
- Y-intercept: (0, 21)

X-intercepts: Set y = 0
- 0 = (x - 5)² - 4
- (x - 5)² = 4
- x - 5 = ±2
- x = 7 or x = 3
- X-intercepts: (3, 0) and (7, 0)

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## Problem 2: y = -(x + 3)² - 1

Vertex: h = -3, k = -1
- Vertex: (-3, -1)

Axis of symmetry: x = -3
- Axis of symmetry: x = -3

Direction of opening: a = -1 (negative)
- Opens downward

Min/Max value: Since it opens downward, there's a maximum
- Maximum value: -1

Y-intercept: Set x = 0
- y = -(0 + 3)² - 1 = -9 - 1 = -10
- Y-intercept: (0, -10)

X-intercepts: Set y = 0
- 0 = -(x + 3)² - 1
- (x + 3)² = -1
- No real solution (can't take square root of negative number)
- X-intercepts: none

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## Problem 3: y = -(x - 1)² + 1

Vertex: h = 1, k = 1
- Vertex: (1, 1)

Axis of symmetry: x = 1
- Axis of symmetry: x = 1

Direction of opening: a = -1 (negative)
- Opens downward

Min/Max value: Maximum
- Maximum value: 1

Y-intercept: Set x = 0
- y = -(0 - 1)² + 1 = -1 + 1 = 0
- Y-intercept: (0, 0)

X-intercepts: Set y = 0
- 0 = -(x - 1)² + 1
- (x - 1)² = 1
- x - 1 = ±1
- x = 2 or x = 0
- X-intercepts: (0, 0) and (2, 0)

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## Problem 4: y = (x + 1)² + 3

Vertex: h = -1, k = 3
- Vertex: (-1, 3)

Axis of symmetry: x = -1
- Axis of symmetry: x = -1

Direction of opening: a = 1 (positive)
- Opens upward

Min/Max value: Minimum
- Minimum value: 3

Y-intercept: Set x = 0
- y = (0 + 1)² + 3 = 1 + 3 = 4
- Y-intercept: (0, 4)

X-intercepts: Set y = 0
- 0 = (x + 1)² + 3
- (x + 1)² = -3
- No real solution
- X-intercepts: none

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## Problem 5: y = -(x + 2)² + 1

Vertex: h = -2, k = 1
- Vertex: (-2, 1)

Axis of symmetry: x = -2
- Axis of symmetry: x = -2

Direction of opening: a = -1 (negative)
- Opens downward

Min/Max value: Maximum
- Maximum value: 1

Y-intercept: Set x = 0
- y = -(0 + 2)² + 1 = -4 + 1 = -3
- Y-intercept: (0, -3)

X-intercepts: Set y = 0
- 0 = -(x + 2)² + 1
- (x + 2)² = 1
- x + 2 = ±1
- x = -1 or x = -3
- X-intercepts: (-1, 0) and (-3, 0)

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## Problem 6: y = -(x + 5)²

Vertex: h = -5, k = 0
- Vertex: (-5, 0)

Axis of symmetry: x = -5
- Axis of symmetry: x = -5

Direction of opening: a = -1 (negative)
- Opens downward

Min/Max value: Maximum
- Maximum value: 0

Y-intercept: Set x = 0
- y = -(0 + 5)² = -25
- Y-intercept: (0, -25)

X-intercepts: Set y = 0
- 0 = -(x + 5)²
- (x + 5)² = 0
- x = -5
- X-intercepts: (-5, 0) (only one, which is also the vertex)

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Final Answer:

Problem 1: Vertex: (5, -4), Axis: x = 5, Opens: up, Min: -4, Y-int: (0, 21), X-ints: (3, 0), (7, 0)

Problem 2: Vertex: (-3, -1), Axis: x = -3, Opens: down, Max: -1, Y-int: (0, -10), X-ints: none

Problem 3: Vertex: (1, 1), Axis: x = 1, Opens: down, Max: 1, Y-int: (0, 0), X-ints: (0, 0), (2, 0)

Problem 4: Vertex: (-1, 3), Axis: x = -1, Opens: up, Min: 3, Y-int: (0, 4), X-ints: none

Problem 5: Vertex: (-2, 1), Axis: x = -2, Opens: down, Max: 1, Y-int: (0, -3), X-ints: (-1, 0), (-3, 0)

Problem 6: Vertex: (-5, 0), Axis: x = -5, Opens: down, Max: 0, Y-int: (0, -25), X-ints: (-5, 0)