Math worksheet for graphing parabolas in standard form with coordinate grids for plotting.

Looking at the worksheet, there are four problems to solve:

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Problem 1:
Write equation in standard form (if not already), then graph and label vertex, axis of symmetry, direction of opening, focus, and directrix.
Equation given: y = x² + 5

This is already in standard form for a vertical parabola: y = ax² + bx + c → here, a=1, b=0, c=5.

Vertex: For y = ax² + bx + c, vertex x-coordinate = -b/(2a) = 0/2 = 0 → y = 0² + 5 = 5 → Vertex: (0, 5)

Axis of symmetry: x = 0 (the y-axis)

Direction of opening: Since a = 1 > 0 → opens upward

Focus and Directrix:
For vertical parabola y = a(x-h)² + k, the distance from vertex to focus is p = 1/(4a)
Here, a = 1 → p = 1/4

Since it opens upward:
- Focus: (h, k + p) = (0, 5 + 0.25) = (0, 5.25)
- Directrix: y = k - p = 5 - 0.25 = y = 4.75

Graph: Plot vertex at (0,5), draw symmetric curve opening up through points like (-1,6), (1,6), etc. Draw dashed line x=0 (axis), mark focus at (0,5.25), draw directrix as horizontal dashed line at y=4.75.

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Problem 2:
Equation: x = 2(y + 3)² - 1

This is a horizontal parabola (x in terms of y). Standard form: x = a(y-k)² + h

Here, a = 2, k = -3, h = -1

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

Axis of symmetry: y = k → y = -3

Direction of opening: a = 2 > 0 → opens right

p = 1/(4a) = 1/(8) = 0.125

Since it opens right:
- Focus: (h + p, k) = (-1 + 0.125, -3) = (-0.875, -3)
- Directrix: x = h - p = -1 - 0.125 = x = -1.125

Graph: Plot vertex at (-1,-3), open right, symmetric over y=-3. Mark focus slightly right of vertex, draw directrix as vertical dashed line left of vertex.

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Problem 3:
Given: vertex (2, 3), focus (5, 3)

Same y-coordinate → horizontal parabola (opens left or right)

Distance from vertex to focus: |5 - 2| = 3 → so p = 3

Since focus is to the right of vertex → opens right

Standard form for horizontal parabola: x = a(y - k)² + h
We know h=2, k=3, and p = 1/(4a) → 3 = 1/(4a) → 4a = 1/3 → a = 1/12

So equation: x = (1/12)(y - 3)² + 2

Directrix: x = h - p = 2 - 3 = x = -1

Graph: Vertex at (2,3), opens right, directrix at x=-1, focus at (5,3).

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Problem 4:
Given: vertex (9, -2), directrix x = 3

Directrix is vertical → parabola opens horizontally.

Vertex x = 9, directrix x = 3 → distance between them = |9 - 3| = 6 → so p = 6

Since directrix is to the left of vertex, parabola opens right

Again, use x = a(y - k)² + h
h = 9, k = -2
p = 1/(4a) → 6 = 1/(4a) → 4a = 1/6 → a = 1/24

Equation: x = (1/24)(y + 2)² + 9

Focus: Since opens right, focus is at (h + p, k) = (9 + 6, -2) = (15, -2)

Graph: Vertex at (9,-2), opens right, directrix at x=3, focus at (15,-2).

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Final Answer:

1. Equation: y = x² + 5; Vertex: (0,5); Axis: x=0; Opens: Up; Focus: (0, 5.25); Directrix: y=4.75
2. Equation: x = 2(y+3)² - 1; Vertex: (-1,-3); Axis: y=-3; Opens: Right; Focus: (-0.875, -3); Directrix: x=-1.125
3. Equation: x = (1/12)(y - 3)² + 2; Directrix: x = -1
4. Equation: x = (1/24)(y + 2)² + 9; Focus: (15, -2)