Problem 11 - Algebra 2 and Trigonometry

Question

\(A(2,7)\) is a fixed point in the coordinate plane. Let \(B(x, 7)\) be any point on the same horizontal line. If \(A B=\mathrm{h}(x),\) express \(\mathrm{h}(x)\) in terms of \(x .\)

Step-by-Step Solution

Answer

\(\mathrm{h}(x) = |x - 2|\)

Step 1: Understand the Problem

We need to express the distance between two points, \(A(2,7)\) and \(B(x, 7)\), in terms of \(x\). The points share the same \(y\)-coordinate.

Step 2: Use the Distance Formula

The distance \(AB\) on the coordinate plane between points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula: \[\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] In this case, since both points have the same \(y\)-coordinate, the formula simplifies due to \(y_2 - y_1 = 0\).

Step 3: Simplify the Expression

Substitute the coordinates of points \(A(2, 7)\) and \(B(x, 7)\) into the simplified distance formula: \[\text{Distance} = \sqrt{(x - 2)^2 + (7 - 7)^2} = \sqrt{(x - 2)^2}\] Since squaring and then taking the square root of a number returns its absolute value, we have:\[|x - 2|\] representing the distance \(AB\).

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional plane with a horizontal axis (x-axis) and a vertical axis (y-axis). It is used to locate points defined by ordered pairs of numbers, known as coordinates, which are written as (x, y)

• The first number in a coordinate pair (x) denotes the horizontal location from the origin (0,0).
• The second number (y) represents the vertical position from the origin.

On the coordinate plane, each axis is made up of evenly spaced intervals, which help in identifying the exact location of points.
In our given exercise, we have two points on the coordinate plane: Point A is located at (2,7) and Point B is at (x,7). These points share the same vertical position because their y-coordinates are equal, making them lie on a horizontal line in the plane. The distance between these points is solely dependent on their x-coordinates.

Absolute Value

Absolute value is a mathematical concept that represents the magnitude or distance of a number from zero on a number line. It is denoted using vertical bars, like |x|

• This essentially means no matter what the number inside the bars, its absolute value is always non-negative.
• It represents distance, and distance cannot be negative.

In the context of our exercise, the distance between points A and B is expressed using absolute value. When calculating the distance as |x - 2|, we take the difference between the x-coordinates of points A (which is 2) and B (which is x), and consider their absolute value. This ensures the distance is always a positive number or zero, simplified to the magnitude of the horizontal shift.

Horizontal Line

A horizontal line on the coordinate plane runs parallel to the x-axis, having the same y-coordinate at every point along its path.
Since there is no change in the vertical position or y-coordinate, this aligns perfectly with our exercise, where points A(2, 7) and B(x, 7) are on the horizontal line y = 7. The key characteristic of a horizontal line is:

• It has a zero slope, meaning it does not rise or fall as it moves horizontally.
• Points along this line maintain constant y-values while x-values can differ, causing only horizontal movement.

Thus, the distance between points on this line is purely dependent on changes in the x-coordinates only, without influence from y-coordinates.

Expressing Distance in Terms of a Variable

When expressing distance between points on a coordinate plane in terms of a variable, we aim to construct a formula that uses variables to represent one or both coordinates. This is especially useful in situations where one point is fixed and the other varies.
In the given problem, point A(2,7) is fixed while point B(x,7) varies along the horizontal line. Here's how we express distance:

• Use the distance formula, but take into account the simplification from identical y-coordinates.
• The result is a distance that centers only around differences in their x-coordinates, leading to |x - 2|.

By expressing in terms of a variable, we achieve a flexible way to determine the distance for any given x, showing its dependency solely on changing horizontal positions.