Question 1

What is a vertex in mathematics?

Accepted Answer

A vertex is the highest or lowest point on a parabola, depending on whether it opens upward or downward. For a quadratic function y = ax² + bx + c, the vertex represents the maximum or minimum value of the function. The vertex coordinates are (h, k) where h = -b/(2a) and k = f(h).

Question 2

How do you find the vertex of a parabola?

Accepted Answer

To find the vertex of a parabola in standard form y = ax² + bx + c: 1) Calculate h = -b/(2a) for the x-coordinate, 2) Substitute h into the equation to find k = f(h) for the y-coordinate, 3) The vertex is (h, k). Alternatively, you can complete the square to convert to vertex form y = a(x - h)² + k where (h, k) is the vertex.

Question 3

What is the axis of symmetry?

Accepted Answer

The axis of symmetry is a vertical line that passes through the vertex of a parabola, dividing it into two mirror-image halves. For a quadratic function y = ax² + bx + c, the axis of symmetry is the line x = -b/(2a). This line represents the x-coordinate of the vertex and is always vertical for standard parabolas.

Question 4

How do you find the focus and directrix of a parabola?

Accepted Answer

For a parabola in vertex form y = a(x - h)² + k: The focus is located at (h, k + 1/(4a)), and the directrix is the horizontal line y = k - 1/(4a). The focus is a point inside the parabola, and the directrix is a line outside it. All points on the parabola are equidistant from the focus and directrix.

Question 5

What is the difference between vertex form and standard form?

Accepted Answer

Vertex form y = a(x - h)² + k directly shows the vertex coordinates (h, k) and is useful for graphing and finding key properties. Standard form y = ax² + bx + c is better for finding roots and general analysis. You can convert between forms using the vertex formula h = -b/(2a) and k = f(h), or by completing the square.

Question 6

How do you convert from standard form to vertex form?

Accepted Answer

To convert y = ax² + bx + c to vertex form: 1) Factor out a from the first two terms, 2) Complete the square by adding and subtracting (b/(2a))², 3) Simplify to get y = a(x - h)² + k where h = -b/(2a) and k = c - b²/(4a). The vertex form makes it easy to identify the vertex (h, k) and other key properties.

Question 7

What are the applications of vertex calculations?

Accepted Answer

Vertex calculations are used in: 1) Physics - finding maximum height in projectile motion, 2) Engineering - optimizing bridge designs and satellite dish focus, 3) Business - finding maximum profit or minimum cost points, 4) Computer graphics - rendering parabolic curves and 3D modeling, 5) Architecture - designing arches and structural elements.

Question 8

How do you find the vertex from factored form?

Accepted Answer

For a parabola in factored form y = a(x - r₁)(x - r₂): 1) The roots are x = r₁ and x = r₂, 2) The x-coordinate of the vertex is h = (r₁ + r₂)/2 (midpoint of the roots), 3) The y-coordinate is k = a(h - r₁)(h - r₂), 4) The vertex is (h, k). This method works because the vertex is always at the midpoint between the roots.

Question 9

What is the discriminant and how does it relate to the vertex?

Accepted Answer

The discriminant is b² - 4ac for a quadratic equation ax² + bx + c = 0. It determines the nature of the roots: positive discriminant means two real roots, zero discriminant means one real root (vertex touches x-axis), and negative discriminant means no real roots. While the discriminant doesn't directly give the vertex, it helps determine if the parabola intersects the x-axis.

Question 10

How do you find the maximum or minimum value using the vertex?

Accepted Answer

The vertex gives the maximum or minimum value of a quadratic function. For y = ax² + bx + c: if a > 0, the vertex is the minimum point, and if a < 0, the vertex is the maximum point. The y-coordinate of the vertex (k) is the maximum or minimum value. This is crucial for optimization problems in various fields like physics, engineering, and economics.

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