Finding the Equation of the Parabola using Calculus

TL;DR

Find the equation of a parabola that passes through a specific point and is tangent to a given line.

Transcript

find the equation of the parabola that passes through the point 0a 5 and is tangent to the line y = 3x - 3 at 1 comma 0 solution let's go ahead and write down everything that we're given so we're told it passes through 0 comma 5 that means that F of 0 is equal to five okay we're also told it's tangent to this line at 1 comma 0 that means that F of ... Read More

Key Insights

• 💁 The general form of the parabola equation is F(x) = ax^2 + bx + 5.
• 😥 The point (0, 5) provides the value of c in the equation.
• 🫥 The condition of being tangent to the line gives the value of b.
• ❓ The derivative of the parabola gives the equation for the slope at (1, 0).
• 😃 Solving the system of equations results in the values of a, b, and c.
• ❓ The final equation of the parabola is F(x) = 8x^2 - 13x + 5.
• ❓ The problem involves using algebra and calculus concepts.

Questions & Answers

Q: What are the conditions given to find the equation of the parabola?

The conditions are that the parabola passes through (0, 5), is tangent to the line y = 3x - 3 at (1, 0), and has a derivative of 3 at x = 1.

Q: How is the value of c determined?

The value of c is found by substituting x = 0 and y = 5 into the general form of the parabola equation, which results in c = 5.

Q: How is the value of b determined?

The value of b is obtained by using the condition that F(1) = 0. Substituting x = 1 into the parabola equation and solving for b gives b = -3.

Q: How are the values of a and b determined?

By setting up a system of equations using the conditions F(1) = 0 and F'(1) = 3, the values of a = 8 and b = -3 are obtained through algebraic manipulation.

Summary & Key Takeaways

• The problem involves finding the equation of a parabola that passes through the point (0, 5) and is tangent to the line y = 3x - 3 at (1, 0).

• By using the conditions provided, the values of a, b, and c can be determined.

• The equation of the parabola is F(x) = 8x^2 - 13x + 5.