Let's carefully examine the expression [tex]\( x^2 + 4 \)[/tex] and understand its components and nature.
1. Understanding the Terms:
- The given expression consists of two terms: [tex]\( x^2 \)[/tex] and [tex]\( 4 \)[/tex].
- [tex]\( x^2 \)[/tex]: This is a quadratic term where [tex]\( x \)[/tex] is a variable, and the exponent 2 indicates that [tex]\( x \)[/tex] is squared.
- [tex]\( 4 \)[/tex]: This is a constant term.
2. Nature of the Expression:
- This type of expression is known as a quadratic expression because the highest power of the variable [tex]\( x \)[/tex] is 2.
3. Graphical Representation:
- If we were to graph the expression [tex]\( y = x^2 + 4 \)[/tex], it would represent a parabola that opens upwards (because the coefficient of [tex]\( x^2 \)[/tex] is positive).
- The vertex of this parabola occurs at the minimum point of the graph. Notice that when [tex]\( x = 0 \)[/tex], [tex]\( y = 4 \)[/tex]. Hence, the vertex of the parabola is at the point [tex]\( (0, 4) \)[/tex].
4. Behavior of the Expression:
- As [tex]\( x \)[/tex] increases or decreases (becomes more positive or more negative), the value of [tex]\( y \)[/tex] increases because [tex]\( x^2 \)[/tex] dominates the constant term [tex]\( 4 \)[/tex].
- Therefore, for large positive or negative values of [tex]\( x \)[/tex], [tex]\( y \)[/tex] approximates [tex]\( x^2 \)[/tex] (shifted 4 units up along the y-axis).
In summary, the expression [tex]\( x^2 + 4 \)[/tex] is a simple quadratic expression with a constant term added to it, representing a parabola with a vertex at [tex]\( (0, 4) \)[/tex] that opens upwards. This expression, given any real number [tex]\( x \)[/tex], will produce a non-negative value for [tex]\( y \)[/tex] with the minimum value of 4 (since the term [tex]\( x^2 \)[/tex] is always non-negative).
