Find Parabola Equation From Vertex And Point Calculator

Parabola Equation Calculator

Enter the vertex (h, k) and another point (x, y) on the parabola to find its equation in vertex and standard forms.

Parameter	Value
Vertex (h, k)
Point (x, y)
Value of 'a'
Focus
Directrix

Summary of Parabola Parameters.

What is a Find Parabola Equation from Vertex and Point Calculator?

A find parabola equation from vertex and point calculator is a tool used to determine the equation of a parabola when you know the coordinates of its vertex (h, k) and at least one other point (x, y) that lies on the parabola. Parabolas are U-shaped curves that can open upwards, downwards, left, or right. This calculator specifically deals with parabolas that open upwards or downwards, represented by the vertex form equation y = a(x – h)² + k.

Anyone studying algebra, pre-calculus, calculus, or physics (e.g., projectile motion) can use this calculator. It's helpful for students to verify their homework, for teachers to create examples, and for engineers or scientists who encounter parabolic shapes in their work.

A common misconception is that any three points define a unique parabola opening up or down. While three non-collinear points define a unique parabola, if one of those points is the vertex, it provides more specific information, allowing us to use the vertex form directly and find the equation with just one additional point.

Find Parabola Equation from Vertex and Point Calculator Formula and Mathematical Explanation

The vertex form of a parabola with a vertical axis of symmetry is:

y = a(x – h)² + k

Where:

• (h, k) are the coordinates of the vertex.
• (x, y) are the coordinates of any other point on the parabola.
• a is a non-zero constant that determines the direction and width of the parabola. If a > 0, the parabola opens upwards; if a < 0, it opens downwards. The larger the absolute value of a, the narrower the parabola.

To find the equation using the vertex (h, k) and another point (x, y), we first need to find the value of a:

1. Substitute the coordinates of the vertex (h, k) and the point (x, y) into the vertex form equation: y = a(x – h)² + k.
2. Rearrange the equation to solve for a:
   y – k = a(x – h)²
   a = (y – k) / (x – h)² (provided x ≠ h)
3. Once a is found, substitute a, h, and k back into the vertex form to get the specific equation of the parabola.

The standard form of a parabola is y = ax² + bx + c. We can get this by expanding the vertex form:
y = a(x² – 2hx + h²) + k
y = ax² – 2ahx + ah² + k
So, b = -2ah and c = ah² + k.

The focus of such a parabola is at (h, k + 1/(4a)) and the directrix is the line y = k – 1/(4a).

Variables Used in the Find Parabola Equation from Vertex and Point Calculator

Variable	Meaning	Unit	Typical Range
h	x-coordinate of the vertex	Units of length	Any real number
k	y-coordinate of the vertex	Units of length	Any real number
x	x-coordinate of the given point	Units of length	Any real number (ideally x ≠ h)
y	y-coordinate of the given point	Units of length	Any real number
a	Coefficient determining parabola's width and direction	Varies	Any non-zero real number
b	Coefficient of x in standard form	Varies	Any real number
c	Constant term in standard form (y-intercept)	Units of length	Any real number

Understanding the variables in the parabola equation.

Practical Examples (Real-World Use Cases)

Example 1: Suspension Bridge Cable

The cable of a suspension bridge often hangs in the shape of a parabola. Suppose the vertex of the parabolic cable is at (0, 10) (meaning 10 meters above the bridge deck at the center), and one of the support towers is 100 meters away from the center, with the cable attaching at a height of 60 meters (point (100, 60)).

• Vertex (h, k) = (0, 10)
• Point (x, y) = (100, 60)

Using the find parabola equation from vertex and point calculator or the formula a = (y – k) / (x – h)²:
a = (60 – 10) / (100 – 0)² = 50 / 10000 = 0.005
The equation is y = 0.005(x – 0)² + 10, or y = 0.005x² + 10.

Example 2: Path of a Projectile

Ignoring air resistance, a projectile follows a parabolic path. If a ball is thrown and reaches its maximum height (vertex) of 15 meters at a horizontal distance of 10 meters from the thrower, and it lands 20 meters away (point (20, 0), assuming ground is y=0 and thrower is at x=0), we have:

• Vertex (h, k) = (10, 15)
• Point (x, y) = (20, 0)

Using the find parabola equation from vertex and point calculator:
a = (0 – 15) / (20 – 10)² = -15 / 100 = -0.15
The equation is y = -0.15(x – 10)² + 15.

How to Use This Find Parabola Equation from Vertex and Point Calculator

1. Enter Vertex Coordinates: Input the x-coordinate (h) and y-coordinate (k) of the parabola's vertex into the "Vertex (h)" and "Vertex (k)" fields, respectively.
2. Enter Point Coordinates: Input the x-coordinate (x) and y-coordinate (y) of the other point that lies on the parabola into the "Point (x)" and "Point (y)" fields.
3. Calculate: The calculator will automatically update the results as you type. If not, click the "Calculate" button.
4. Read Results:
   • Primary Result: Shows the equation of the parabola in vertex form y = a(x – h)² + k with the calculated value of a.
   • Intermediate Results: Displays the value of a, the equation in standard form y = ax² + bx + c, the coordinates of the focus, and the equation of the directrix.
   • Graph: A visual representation of the parabola, vertex, point, focus, and directrix is shown.
   • Table: Summarizes the key parameters.
5. Reset: Click "Reset" to clear the fields and start over with default values.
6. Copy Results: Click "Copy Results" to copy the main equation, 'a' value, focus, and directrix to your clipboard.

If the x-coordinate of the point is the same as the x-coordinate of the vertex, and the y-coordinates differ, no such function y=a(x-h)²+k exists. If the point is the vertex, 'a' is indeterminate.

Key Factors That Affect Parabola Equation Results

1. Vertex Coordinates (h, k): The vertex is the turning point of the parabola and directly defines the h and k values in the vertex form y = a(x – h)² + k. Changing the vertex shifts the parabola horizontally and vertically.
2. Point Coordinates (x, y): The other point on the parabola is crucial for determining the value of a. The position of this point relative to the vertex dictates how wide or narrow the parabola is and whether it opens upwards or downwards.
3. Difference (x – h): The horizontal distance between the point and the vertex. If this is zero, and y ≠ k, a parabola of the form y=a(x-h)²+k is not possible. The square of this difference significantly influences 'a'.
4. Difference (y – k): The vertical distance between the point and the vertex. This, divided by (x – h)², gives the value of a.
5. Value of 'a': This coefficient, derived from the vertex and the point, determines the parabola's steepness and direction. A positive 'a' means it opens up, negative 'a' means down. Larger |a| means narrower.
6. Axis of Symmetry: For y = a(x – h)² + k, the axis of symmetry is always the vertical line x = h, passing through the vertex. The focus also lies on this axis.

Frequently Asked Questions (FAQ)

What is the vertex form of a parabola?

The vertex form is y = a(x – h)² + k, where (h, k) is the vertex and a is a constant.

What if the given point is the vertex?

If the point (x, y) is the same as the vertex (h, k), then x-h=0 and y-k=0. The formula for 'a' becomes 0/0, which is indeterminate. This means infinitely many parabolas can have that vertex and pass through it (as the point is the vertex itself). You need a point *other than* the vertex to uniquely define 'a' for a specific parabola.

What if the x-coordinate of the point is the same as the vertex (x=h), but y is different?

If x=h and y ≠ k, then (x-h)² = 0, and y-k ≠ 0. The formula for 'a' would involve division by zero, meaning 'a' is undefined. A parabola of the form y=a(x-h)²+k (opening up or down) cannot pass through two distinct points with the same x-coordinate, one of which is the vertex. This would imply a vertical line, not a function of x in this form.

How does the value of 'a' affect the parabola?

If 'a' is positive, the parabola opens upwards. If 'a' is negative, it opens downwards. The larger the absolute value of 'a', the narrower (steeper) the parabola. The smaller the absolute value of 'a', the wider it is.

Can I find the equation of a parabola that opens sideways using this calculator?

No, this find parabola equation from vertex and point calculator is specifically for parabolas that open upwards or downwards, represented by y = a(x – h)² + k. For parabolas opening sideways, the form is x = a(y – k)² + h.

What is the focus of a parabola?

The focus is a point inside the parabola on its axis of symmetry. For y = a(x – h)² + k, the focus is at (h, k + 1/(4a)).

What is the directrix of a parabola?

The directrix is a line perpendicular to the axis of symmetry, outside the parabola. For y = a(x – h)² + k, the directrix is the line y = k – 1/(4a). Every point on the parabola is equidistant from the focus and the directrix.

How do I find the x-intercepts or y-intercept?

To find the y-intercept, set x=0 in the equation y = ax² + bx + c, so y=c. To find the x-intercepts (roots), set y=0 and solve the quadratic equation ax² + bx + c = 0 using the quadratic formula or factoring. You can use a quadratic equation solver for this.

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