Problem 70 - Calculus Single Variable

Question

The graph of \(y=x^{2}\) is said to be concave up. One way to define this property is as follows: If \(P=\left(a, a^{2}\right)\) and \(Q=\) \(\left(b, b^{2}\right)\) are any two points on the graph of \(y=x^{2},\) then the line segment \(\overline{P Q}\) lies above the graph of \(y=x^{2} .\) What is the equation of the line that passes through \(P\) and \(Q ?\) Algebraically verify that the midpoint of \(\overline{P Q}\) lies above the parabola.

Step-by-Step Solution

Answer

Line equation: \(y = (b+a)x - a(b+a) + a^2\). Midpoint is above parabola as \(\frac{a^2 + b^2}{2} > \frac{(a+b)^2}{4}\).

Step 1: Find the Slope of Line PQ

Given points \(P(a, a^2)\) and \(Q(b, b^2)\), the slope \(m\) of the line segment \(\overline{PQ}\) can be found using the formula for the slope: \[ m = \frac{b^2 - a^2}{b-a}. \]Expand this to get \( m = b+a \) since \( (b^2 - a^2) = (b-a)(b+a) \).

Step 2: Define the Equation of Line PQ

The equation of a line in point-slope form is \(y - y_1 = m(x - x_1)\). Use point \(P(a, a^2)\) and slope \(b + a\):\[y - a^2 = (b + a)(x - a).\] Simplify to get the line equation:\[ y = (b + a)x - a(b + a) + a^2. \]

Step 3: Find the Midpoint of PQ

The midpoint \(M\) of line segment \(\overline{PQ}\) is given by:\[ M\left( \frac{a+b}{2}, \frac{a^2+b^2}{2} \right). \]

Step 4: Determine the y-coordinate of Midpoint M on Parabola

Substitute \(x = \frac{a+b}{2}\) into the equation of the parabola \(y = x^2\):\[ y = \left( \frac{a+b}{2} \right)^2 = \frac{(a+b)^2}{4}. \]

Step 5: Compare Midpoint's y-coordinate with Parabola

Compute \( (a^2 + b^2)/2 \) and compare it with \( (a+b)^2/4 \):\[ \frac{a^2 + b^2}{2} \] and \[ \frac{(a+b)^2}{4} = \frac{a^2 + 2ab + b^2}{4}. \]Simplifying the comparison, \[ \frac{a^2 + b^2}{2} = \frac{2a^2 + 2b^2}{4} > \frac{a^2 + 2ab + b^2}{4}\] due to the positive term \(2ab\) implying the midpoint is above the parabola.

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness, describing how the line tilts from the horizontal axis. To find the slope between two points, you use their coordinates. Suppose we have two points, \(P(a, a^2)\) and \(Q(b, b^2)\) on a graph. The formula you will use to determine the slope \(m\) of the line that connects these two points is:

• \( m = \frac{b^2 - a^2}{b-a} \)

Notice how the formula represents the change in \(y\) values relative to the change in \(x\) values. We simplify \(b^2 - a^2\) using the difference of squares, leading to \( m = b + a \), which implies that the difference in y-values also considers how far apart the \(x\) values are. This information is foundational to drawing and understanding lines in coordinate geometry.

Midpoint Formula

Finding the midpoint between two points is like finding the average position between them on a line segment. It's calculated using their respective coordinates. For points \(P(a, a^2)\) and \(Q(b, b^2)\), the midpoint \(M\) is found using:

• \( M\left( \frac{a+b}{2}, \frac{a^2+b^2}{2} \right) \)

This formula takes pairwise averages of the coordinates to compute the midpoint, effectively determining a new point halfway along the line segment connecting \(P\) and \(Q\). When verifying the position of this midpoint against a curve like a parabola, it helps to determine whether the midpoint lies above, below, or on the curve. For parabolas, like \(y = x^2\), computing these comparisons provides insight into the shape and form of the parabola.

Parabola

A parabola is a U-shaped curve on a graph described by a quadratic equation like \(y = x^2\). It is remarkably symmetric, centered around a vertical line called the axis of symmetry. Parabolas can either open upwards or downwards depending on the coefficient of \(x^2\). For \(y = x^2\), the parabola is said to be "concave up" as it hollowly faces upwards.

• The turning point is its vertex, located at the origin \((0,0)\) for \(y = x^2\).
• As \(x\) values increase positively or negatively, the \(y\) always increases, signifying uplift.
• The parabola's shape and position can be affected by changing coefficients and constants.

Used frequently in physics to describe trajectories and in algebra to demonstrate fundamental principles of equations, parabolas' artistic symmetry is also a feature engineers take advantage of in designs from satellite dishes to bridges.

Point-Slope Form

The point-slope form is a common method for writing the equation of a line, especially when you know the slope of a line and a point on it. The general form is given by:

• \(y - y_1 = m(x - x_1)\)

Where \((x_1, y_1)\) represents the coordinates of a known point on the line, and \(m\) is the slope. This form is particularly useful for constructing the line equation quickly from partial information. Applying it to the problem, if we know the point \(P(a, a^2)\) and calculated the slope as \(b + a\), the line's equation can be crafted as:

• \(y - a^2 = (b + a)(x - a)\)

This can further be simplified for easier analysis and coverage of the line across the plane. It allows us to substitute values and verify specific positions relative to other objects, such as parabolas, to help illustrate concepts of overlying line segments and curves.