Problem 68 - Precalculus with Limits

Question

In Exercises 65-68, find an equation of the tangent line to the parabola at the given point, and find the \(x\)-intercept of the line. \(y=-2x^2, (2, -8)\)

Step-by-Step Solution

Answer

The equation of the tangent line to the parabola at the point (2, -8) is \(y = -8x + 8\) and the \(x\)-intercept of the line is 1.

Step 1: Find the derivative of the function

The derivative of the given function \(y = -2x^2\) is \(-4x\) using power rule.

Step 2: Find the slope of the tangent line

The slope of the tangent line on the function \(y = -2x^2\) at the point (2, -8) is calculated by substituting the \(x\) value of the point into the derivative, such that the slope, \(m = -4(2) = -8\).

Step 3: Find the equation of the tangent line

The formula for a linear equation is \(y = mx + c\). Here, \(m = -8\) (from Step 2) and the point (2, -8) can be substituted in to solve for the intercept \(c\). From this, \(c = -8 - (-8*2) = 8\), thus the equation of the tangent line is \(y = -8x + 8\).

Step 4: Find the x-intercept of the line

The \(x\)-intercept of the line is found by setting \(y = 0\) in the equation of the line, thus \(0 = -8x + 8 \Rightarrow x = 1\).

Key Concepts

Understanding Derivatives

When dealing with problems involving tangent lines to curves, the term "derivative" becomes crucial. The derivative of a function is a fundamental concept in calculus that represents the rate of change of the function's value with respect to its input. In simpler terms, the derivative tells you how a function is changing at any given point.

For example, if you have a function like \( y = -2x^2 \), the derivative using the power rule is calculated by multiplying the exponent by the coefficient in front of \( x \), and then subtracting one from the exponent. This results in the derivative \( -4x \).

• The derivative shows how the function \( y = -2x^2 \) is changing as \( x \) changes.
• It helps in finding the slope of the tangent line at any given point on the curve.

Recognizing the derivative's role is vital as it helps find important features of the graph, such as slopes of tangent lines.

Slopes of Tangent Lines

The slope of a tangent line at a specific point on a curve provides important information about how steep the line is and in which direction it goes.

To find the slope of the tangent line at a specific point, we evaluate the derivative at that point.

• In our problem, the tangent line needs to be found at the point \((2, -8)\).
• The given derivative of the function \( y = -2x^2 \) is \(-4x\).
• Substituting \( x = 2 \) into \(-4x\) gives us the slope \( -8 \).

This tells us that at \( x = 2 \), the tangent line has a negative slope, specifically -8. It indicates that the line decreases steeply as it moves from left to right.

Graphs of Parabolas

Understanding the shape of a parabola helps in visualizing how tangent lines coincide with parabolas. A parabola is a U-shaped curve that can open upward or downward.

• The parabola described by \( y = -2x^2 \) opens downward because of the negative coefficient of \( x^2 \).
• The point \((2, -8)\) lies on the curve, and our task was to find the tangent at this point.

Parabolas have symmetric properties, and the vertex, which is the highest or lowest point, is crucial in understanding its geometry.

Tangent lines intersect the curve at exactly one point, the point of tangency, and understanding this helps in visualizing how the slope and the derivative come into play.

Finding the X-Intercept

Once the equation of a tangent line is found, we can determine its x-intercept. The x-intercept is where the line crosses the x-axis.

• For a line described by \( y = -8x + 8 \), the x-intercept is found by setting \( y = 0 \).
• Solving \( 0 = -8x + 8 \) yields \( x = 1 \).

This point on the x-axis is where the tangent line passes through as it cuts across from one side of the x-axis to the other.

Identifying x-intercepts is useful for understanding where a function or its tangent intersects the axis and typically involves solving simple algebraic equations.