Analysis of Quadratic Coordinate Data on the Cartesian Plane

Overview of the Mathematical Visual Source

The provided data originates from BYJU'S The Learning App, representing a mathematical visualization of a specific numeric relationship on a Cartesian coordinate system. The visual consists of an $x$-axis and a $y$-axis, featuring several discrete points that, when connected, form a smooth parabolic curve. This type of graph is typical in algebra and calculus for demonstrating the behavior of quadratic functions through a graphical representation of coordinate pairs.

Detailed Analysis of the Coordinate Ticks and Scaling

The horizontal axis, known as the $x$-axis, spans a visible range from roughly $-5$ to $5$. The vertical axis, known as the $y$-axis, displays increments of $5$ units to represent the magnitude of the dependent variable. Starting from the origin where the axes intersect, the explicitly labeled increments on the $y$-axis are $5$, $10$, $15$, and $20$. The graph extends beyond the $20$ mark, as indicated by the label $20+$ at the top of the vertical axis. On the negative side of the vertical axis, a tick mark is present at $-5$, though no data points are plotted in that negative region. The precise grid system allows for the accurate placement of mathematical values.

Comprehensive List of Characterized Coordinate Points

The graph identifies seven unique coordinates that define the shape and position of the curve. These points are labeled to provide exact data values for the underlying function. Starting from the leftmost point and moving rightward across the plot, the coordinates are as follows:

1. $(-3, 16)$: Located in the second quadrant, this point shows that when the independent variable $x$ is equal to $-3$, the dependent variable $y$ is $16$.
2. $(-2, 7)$: As the $x$ value increases toward zero, reaching $-2$, the value of $y$ decreases to a value of $7$.
3. $(-1, 2)$: Further to the right, when $x$ is exactly $-1$, the corresponding output value on the vertical axis is $2$.
4. $(-0.25, 0.875)$: This specific point is positioned very close to the vertical axis and represents the local minimum of the graph. At $x = -0.25$, the $y$ value is calculated as exactly $0.875$. This suggests a vertex located in the second quadrant.
5. $(1, 4)$: Transitioning into the first quadrant, when $x$ takes the positive value of $1$, the output result for $y$ is $4$.
6. $(2, 11)$: For an $x$ value of $2$, the graph rises significantly to a $y$ value of $11$.
7. $(3, 22)$: The final labeled point on the right side of the graph illustrates that when $x$ reaches $3$, the $y$ value reaches its highest visible point on this plot, which is $22$.

Derivation of the Mathematical Function

The arrangement of these plotted points follows a quadratic pattern, which is mathematically modeled by the general equation $y = ax^2 + bx + c$. By utilizing the provided coordinates from the source, such as $(1, 4)$, $(2, 11)$, and $(3, 22)$, a system of linear equations can be established to solve for the unknown coefficients $a$, $b$, and $c$.

Substituting the point $(1, 4)$ yields the equation: $a(1)^2 + b(1) + c = 4$ which simplifies to $a + b + c = 4$

Substituting the point $(2, 11)$ yields the equation: $a(2)^2 + b(2) + c = 11$ which simplifies to $4a + 2b + c = 11$

Substituting the point $(3, 22)$ yields the equation: $a(3)^2 + b(3) + c = 22$ which simplifies to $9a + 3b + c = 22$

By solving this system, we find that the coefficients are $a = 2$, $b = 1$, and $c = 1$. Therefore, the definitive function represented by the graphical data is: $y = 2x^2 + x + 1$

Verification and Geometric Properties of the Curve

The derived function $y = 2x^2 + x + 1$ can be verified against the identified vertex point $(-0.25, 0.875)$ found in the original visual. Using the standard formula for the vertex $x$-coordinate, $x = \frac{-b}{2a}$, we find: $x = \frac{-1}{2 \times 2} = -0.25$

The corresponding $y$-value for the vertex is found by substituting this back into the equation: $y = 2(-0.25)^2 + (-0.25) + 1$$y = 2(0.0625) - 0.25 + 1$$y = 0.125 - 0.25 + 1 = 0.875$

Geometrically, the positive coefficient $a = 2$ confirms that the parabola opens upwards. The curve exhibits symmetry about the axis of symmetry defined by the vertical line $x = -0.25$. The $y$-intercept of the function is located at $(0, 1)$, occurring where the curve crosses the vertical axis between the points $(-0.25, 0.875)$ and $(1, 4)$.