Question

Find the vertex of the parabola that goes through the points (3,0), (−4,0), and (6,−6).

a) (3, 0)
b) (−4, 0)
c) (6, −6)
d) No vertex exists

Answer 1

The vertex of the parabola is (-3, 0).

Step-by-step explanation:

The vertex of a parabola can be found using the formula x = -b/2a. In this case, the quadratic equation representing the parabola is ax^2 + bx + c = 0. To find the vertex, we need to find the x-coordinate, which is determined by the formula -b/2a. Let's substitute the values from the given points into the formula:

x = (-b) / (2a) = (0 + 0 + (-6)) / (2*1) = -6/2 = -3.

Therefore, the vertex of the parabola is (-3, 0).

A parabola is a type of curve that is defined by a quadratic equation of the form

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y=ax

2

+bx+c, where

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a,

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b, and

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c are constants. The basic shape of a parabola is that of a symmetric curve, and it can open upward or downward.

The standard form of a parabola, when the vertex is at the origin (0,0), is

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=

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2

y=ax

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. This parabola opens upward if

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a>0 and downward if

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a<0.

If the vertex of the parabola is at the point

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(h,k), then the standard form becomes

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(

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y=a(x−h)

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+k. This form allows you to easily determine the direction of opening (upward or downward) and the position of the vertex.

Parabolas have several important properties:

Vertex: The vertex is the point where the parabola reaches its minimum or maximum value. For a parabola in the form

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2

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+

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y=ax

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and

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k=f(h), where

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f(x)=ax

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Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetric halves.

Focus and Directrix (for a parabola in standard form): A parabola can be defined in terms of its focus and directrix. The focus is a point inside the parabola, and the directrix is a line outside the parabola. The distance from any point on the parabola to the focus is equal to the perpendicular distance from that point to the directrix.

Focal Length: The focal length is the distance between the vertex and the focus of the parabola.

Parabolas are commonly encountered in mathematics and physics. They can describe the trajectory of a projectile, the shape of satellite dishes, and various other natural and man-made phenomena.

Answer 2

The answer of the given statement that "the vertex of the parabola that goes through the points (3,0), (−4,0), and (6,−6)" is b) (−4, 0)

Step-by-step explanation:

To find the vertex of the parabola, we can start by recognizing that the vertex lies on the axis of symmetry.

Given that the parabola passes through the points (3,0) and (−4,0), the axis of symmetry is the line
`x = (3 + (-4)) / 2 = -0.5.`

Now, since the vertex lies on this axis, its x-coordinate is -0.5. To find the corresponding y-coordinate, we can use one of the given points.

Let's use (3,0):

The parabola's equation can be expressed in the form
`\(y = a(x - h)^2 + k\)`, where (h, k) is the vertex. Since the axis of symmetry is x = -0.5, the equation becomes
`\(y = a(x + 0.5)^2 + k\).`

Plugging in the coordinates of the point (3,0):

`\[0 = a(3 + 0.5)^2 + k\]`

Solving for k, we find that
`\(k = -4\).`Therefore, the vertex is (-0.5, -4), and the answer is (b) (-4, 0).