Equation in the Cartesian plane
The phrase “graph the following equation in a rectangular coordinate system” refers to plotting the set of all points \((x,y)\) satisfying a given relationship on the Cartesian plane. A specific equation is required, and the representative algebra example used here is \( y = x^2 - 4x + 3 \).
Algebraic form and key features
The quadratic \( y = x^2 - 4x + 3 \) defines a parabola. Completing the square gives a vertex form that displays the minimum immediately:
\[ y = x^2 - 4x + 3 = (x^2 - 4x + 4) - 1 = (x - 2)^2 - 1. \]
Vertex: \((2,-1)\). Axis of symmetry: \(x = 2\). Opening direction: upward (positive coefficient of \(x^2\)). Range: \(y \ge -1\).
Intercepts and a plotting table
The x-intercepts satisfy \(y = 0\), so \(x^2 - 4x + 3 = 0\). Factoring yields:
\[ x^2 - 4x + 3 = (x - 1)(x - 3), \quad \text{so } x = 1 \text{ or } x = 3. \]
The y-intercept satisfies \(x = 0\), so \(y = 3\) and the intercept is \((0,3)\).
| x | \(y = x^2 - 4x + 3\) |
|---|---|
| -1 | \(8\) |
| 0 | \(3\) |
| 1 | \(0\) |
| 2 | \(-1\) |
| 3 | \(0\) |
| 4 | \(3\) |
| 5 | \(8\) |
Graph in a rectangular coordinate system
Consistency properties
Symmetry about \(x = 2\) implies equal y-values for inputs equally spaced from 2, such as \(y(1) = y(3)\) and \(y(0) = y(4)\). The upward opening and the vertex form \(y = (x - 2)^2 - 1\) imply that all plotted points lie on or above \(y = -1\), with the minimum occurring at \((2,-1)\).