Problem 60 - Precalculus

Question

Find the equation of the tangent line to the parabola at the given point. $$y=-2 x^{2},(2,-8)$$

Step-by-Step Solution

Answer

The equation of the tangent line to the parabola \(y=-2x^2\) at the point (2,-8) is \(y = -8x + 8\).

Step 1: Compute the derivative

First, take the derivative of \(y=-2x^2\) the original function. Using the power rule, the derivative of \(y=-2x^2\) is \(-4x\).

Step 2: Find the slope of the tangent line

Next, plug in the x-value of the point of tangency into the derivative to find the slope of the tangent line at that point. The x-value of the point (2,-8) is 2. So, \(-4*2 = -8\).

Step 3: Use the point-slope equation of a line to find the equation

Now that you have the slope of the tangent line and the point of tangency, substitute these values into the point-slope form equation to find the equation of the tangent line. Doing so gives you \(y - (-8) = -8*(x - 2)\). After simplifying the equation, you get \(y = -8x + 8\).

Key Concepts

Derivative

Understanding the derivative is crucial when dealing with calculus, especially when finding the equation of a tangent line. A derivative represents the rate at which a function is changing at any given point, commonly referred to as the slope of the tangent line at that point. To calculate the derivative, you take the original function and apply differentiation rules.

In our exercise, the function is a parabola described by the equation y = -2x^2. To find the derivative, which is the slope of the tangent line at a specific point, you simply apply the rules of differentiation to the function.

Power Rule

One of the most commonly used differentiation rules in calculus is the power rule. This rule allows us to find the derivative of a function that has the form of a power of x, such as x^n. The power rule states that the derivative of x^n is n*x^(n-1).

Applying this rule to our function, y = -2x^2, the '2' in x^2 is our power (n). Using the power rule, we multiply the power by the coefficient, resulting in -4x. This gives us the slope of the tangent line at any point x on the parabola.

Point-Slope Form

The point-slope form is an algebraic equation that describes a straight line with a given slope and passing through a given point. The generic form is y - y1 = m(x - x1), where (x1, y1) is the point the line goes through, and m is the slope of the line.

To find the equation of the tangent line in our exercise, we need the slope from the derivative at the x-coordinate of our given point and the coordinates of the point itself. Plugging these into the point-slope form equation will give us the tangent line equation we're looking for.

Parabola

A parabola is a specific type of curve on a graph. Parabolas represent quadratic functions and have a distinct U-shape, opening either upward or downward depending on the sign of their leading coefficient. The equation of a parabola is generally written as y = ax^2 + bx + c, where a, b, and c are constants.

In our context, the function y = -2x^2 represents a parabola that opens downward, since the coefficient of x^2 is negative. The point of tangency given in the exercise, (2, -8), lies on this parabola. The tangent line to the parabola at this point will touch the curve in exactly one place, without crossing it, emphasizing the importance of the point in finding our tangent line equation.