# Finding the equation for the parabola

• Burnstreak
In summary: Using the original equation, y= ax^2+ bx, when x= 2, y= a(4)+ b= 4a+ b= 2. You now have two equations to solve for a and b.In summary, the problem is to find the parabola of the form y=ax^2+bx that has a tangent with equation y=5x-8 at the point P=(2,2). The equations involved are y=ax^2+bx for the parabola and y-y1=m(x-x1) for the slope. To solve the problem, the derivative of the parabola equation is taken and set equal to the slope, resulting in 5=4a+b
Burnstreak

## Homework Statement

Find the parabola of the form y=ax^2+bx, whose tangent at given point P has equation y.
y=5x-8
P=(2,2)

## Homework Equations

I guess the equations involved would be the equation for the parabola: y=ax^2+bx, and the y-y1=m(x-x1) for the slope.

## The Attempt at a Solution

Using the parabola equation, I found a general derivative and then input the x and y' values in order to try and find a and b, but I was unable to solve for either.

Why in the world do you tell us that you did all those things without showing us exactly what you got when you did it? We can't tell where you went wrong if you don't show us what you did!

You know, I hope, that the line y= 5x- 8 has slope 5 and if x= 2, y= 2.

Okay, well in order to find the slope, I took the derivative of the parabola equation and got 2ax+b for the derivative. I sent it equal to 5 since that is the slope and put in 2 for x since it is given getting 5=4a+b. This is where I got lost. I didn't know what to do once I got to this point.

You also know that when x = 2, y = 2. This gives you another equation.

## What is a parabola?

A parabola is a U-shaped curve that is defined by a quadratic equation. It is a symmetrical shape with a highest or lowest point called the vertex.

## How do you find the equation for a parabola?

The equation for a parabola can be found by using the standard form, y = ax^2 + bx + c, where a, b, and c are constants. These constants can be determined by identifying the coordinates of the vertex and one other point on the parabola.

## What is the significance of the coefficient a in the parabola equation?

The coefficient a represents the slope or steepness of the parabola. If a is positive, the parabola opens upward, and if a is negative, the parabola opens downward. The magnitude of a also determines how wide or narrow the parabola is. A larger absolute value of a results in a narrower parabola.

## Can the equation for a parabola have fractions or decimals?

Yes, the coefficients a, b, and c in the parabola equation can be fractions or decimals. This allows for more precise representations of parabolas with non-integer values.

## Are there other forms of the parabola equation?

Yes, there are other forms of the parabola equation, including the vertex form, y = a(x - h)^2 + k, and the factored form, y = a(x - r)(x - s). Each form has its own advantages and can be useful in different situations. It is important to be familiar with all forms in order to fully understand the properties of parabolas.

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