To determine whether the equation [tex]\( y = x^2 + 2 \)[/tex] is linear or non-linear, let's analyze its structure and properties.
1. Understanding Linear Equations:
- A linear equation in two variables (e.g., [tex]\( x \)[/tex] and [tex]\( y \)[/tex]) can be written in the general form [tex]\( y = mx + b \)[/tex], where:
- [tex]\( y \)[/tex] is the dependent variable.
- [tex]\( x \)[/tex] is the independent variable.
- [tex]\( m \)[/tex] is the slope of the line.
- [tex]\( b \)[/tex] is the y-intercept.
- The key characteristic of a linear equation is that it forms a straight line when graphed on a coordinate plane. Additionally, in a linear equation, the highest power of the variable [tex]\( x \)[/tex] is 1.
2. Analyzing the Given Equation [tex]\( y = x^2 + 2 \)[/tex]:
- Observe that the equation includes a term [tex]\( x^2 \)[/tex], which means that the variable [tex]\( x \)[/tex] is raised to the power of 2.
- The presence of [tex]\( x^2 \)[/tex] indicates that the equation involves a quadratic term rather than a linear one.
3. Conclusion:
- Since the equation contains a quadratic term ([tex]\( x^2 \)[/tex]), it does not fit the form of [tex]\( y = mx + b \)[/tex], and thus, it is not a linear equation.
- Instead, the equation [tex]\( y = x^2 + 2 \)[/tex] describes a parabola when graphed, which is a characteristic of non-linear equations.
Therefore, the equation [tex]\( y = x^2 + 2 \)[/tex] is non-linear.
