Q: What is the focus of a parabola?

The focus is a fixed point inside the parabola from which every point on the parabola is equidistant to the directrix. For a parabola with vertex at (h, k) opening vertically, the focus is at (h, k + p) where p = 1/(4a). The focus lies on the axis of symmetry at distance |p| from the vertex. It's crucial for understanding parabola properties and applications like satellite dishes.

Q: What is the directrix of a parabola?

The directrix is a fixed line perpendicular to the axis of symmetry. Every point on the parabola is equidistant from the focus and the directrix. For a vertical parabola with vertex (h, k), the directrix is the line y = k - p, where p = 1/(4a). The directrix lies on the opposite side of the vertex from the focus, also at distance |p| from the vertex.

Q: How to calculate the axis of symmetry of a parabola?

The axis of symmetry is a vertical line passing through the vertex. To find it: 1) For y = ax² + bx + c, the axis of symmetry is x = -b/(2a). 2) For y = a(x-h)² + k, the axis of symmetry is x = h. 3) For y = ax², the axis of symmetry is x = 0 (the y-axis). This line divides the parabola into two mirror images.

Q: What is the p-value in a parabola?

The p-value (focal parameter) is the distance from the vertex to the focus and also from the vertex to the directrix. For a parabola y = ax², p = 1/(4a). If p > 0, the parabola opens upward or rightward. If p < 0, it opens downward or leftward. The larger |p|, the wider the parabola. This value is essential for understanding the parabola's shape and orientation.

Q: How do you convert between parabola forms?

To convert: Standard to Vertex: Complete the square on y = ax² + bx + c to get y = a(x-h)² + k. Vertex to Standard: Expand y = a(x-h)² + k to get y = ax² + bx + c. General to Standard: Factor out 'a' and complete the square. Each form has advantages: vertex form shows the vertex directly, standard form shows y-intercept, and general form is easy to compute.

Q: What determines if a parabola opens up or down?

The sign of the coefficient 'a' in y = ax² determines opening direction: If a > 0, the parabola opens upward (U-shaped) with a minimum at the vertex. If a 0 it opens right, if a < 0 it opens left. The absolute value |a| affects width: larger |a| makes it narrower.

Q: How are parabolas used in real life?

Parabolas appear in: 1) Projectile motion - objects thrown follow parabolic paths. 2) Satellite dishes - parabolic shape focuses signals to a receiver at the focus. 3) Bridges - suspension bridge cables form parabolic curves. 4) Car headlights - parabolic reflectors direct light beams. 5) Solar cookers - concentrate sunlight at the focus. 6) Architecture - parabolic arches distribute weight efficiently.

Q: What is the relationship between vertex and focus?

The vertex and focus are connected by the axis of symmetry. The focus lies at distance p from the vertex along this axis, where p = 1/(4a). For upward-opening parabolas, the focus is above the vertex. For downward-opening, it's below. The distance |p| affects the parabola's width: smaller |p| means a steeper, narrower parabola. Every point on the parabola is equidistant from the focus and directrix.

Question 1

What is a parabola?

Accepted Answer

A parabola is a U-shaped curve that is the graph of a quadratic function. It's defined as the set of all points equidistant from a fixed point (focus) and a fixed line (directrix). Parabolas appear in physics (projectile motion), engineering (satellite dishes, bridges), and algebra (quadratic equations). The general form is y = ax² + bx + c or in vertex form y = a(x-h)² + k.

Question 2

How do you find the vertex of a parabola?

Accepted Answer

To find the vertex: 1) For standard form y = ax² + bx + c, use x = -b/(2a) to find the x-coordinate, then substitute to find y. 2) For vertex form y = a(x-h)² + k, the vertex is (h, k). 3) For y = ax², the vertex is at the origin (0, 0). The vertex is the parabola's minimum point (if a > 0) or maximum point (if a < 0).

Question 3

What is the focus of a parabola?

Accepted Answer

The focus is a fixed point inside the parabola from which every point on the parabola is equidistant to the directrix. For a parabola with vertex at (h, k) opening vertically, the focus is at (h, k + p) where p = 1/(4a). The focus lies on the axis of symmetry at distance |p| from the vertex. It's crucial for understanding parabola properties and applications like satellite dishes.

Question 4

What is the directrix of a parabola?

Accepted Answer

The directrix is a fixed line perpendicular to the axis of symmetry. Every point on the parabola is equidistant from the focus and the directrix. For a vertical parabola with vertex (h, k), the directrix is the line y = k - p, where p = 1/(4a). The directrix lies on the opposite side of the vertex from the focus, also at distance |p| from the vertex.

Question 5

How to calculate the axis of symmetry of a parabola?

Accepted Answer

The axis of symmetry is a vertical line passing through the vertex. To find it: 1) For y = ax² + bx + c, the axis of symmetry is x = -b/(2a). 2) For y = a(x-h)² + k, the axis of symmetry is x = h. 3) For y = ax², the axis of symmetry is x = 0 (the y-axis). This line divides the parabola into two mirror images.

Question 6

What is the p-value in a parabola?

Accepted Answer

The p-value (focal parameter) is the distance from the vertex to the focus and also from the vertex to the directrix. For a parabola y = ax², p = 1/(4a). If p > 0, the parabola opens upward or rightward. If p < 0, it opens downward or leftward. The larger |p|, the wider the parabola. This value is essential for understanding the parabola's shape and orientation.

Question 7

How do you convert between parabola forms?

Accepted Answer

To convert: Standard to Vertex: Complete the square on y = ax² + bx + c to get y = a(x-h)² + k. Vertex to Standard: Expand y = a(x-h)² + k to get y = ax² + bx + c. General to Standard: Factor out 'a' and complete the square. Each form has advantages: vertex form shows the vertex directly, standard form shows y-intercept, and general form is easy to compute.

Question 8

What determines if a parabola opens up or down?

Accepted Answer

The sign of the coefficient 'a' in y = ax² determines opening direction: If a > 0, the parabola opens upward (U-shaped) with a minimum at the vertex. If a < 0, the parabola opens downward (∩-shaped) with a maximum at the vertex. For horizontal parabolas x = ay², if a > 0 it opens right, if a < 0 it opens left. The absolute value |a| affects width: larger |a| makes it narrower.

Question 9

How are parabolas used in real life?

Accepted Answer

Parabolas appear in: 1) Projectile motion - objects thrown follow parabolic paths. 2) Satellite dishes - parabolic shape focuses signals to a receiver at the focus. 3) Bridges - suspension bridge cables form parabolic curves. 4) Car headlights - parabolic reflectors direct light beams. 5) Solar cookers - concentrate sunlight at the focus. 6) Architecture - parabolic arches distribute weight efficiently.

Question 10

What is the relationship between vertex and focus?

Accepted Answer

The vertex and focus are connected by the axis of symmetry. The focus lies at distance p from the vertex along this axis, where p = 1/(4a). For upward-opening parabolas, the focus is above the vertex. For downward-opening, it's below. The distance |p| affects the parabola's width: smaller |p| means a steeper, narrower parabola. Every point on the parabola is equidistant from the focus and directrix.

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