To determine which of the given options represents a quadratic function, we need to understand what a quadratic function looks like.
A quadratic function is a polynomial function of degree 2, and it can be generally written in the form:
[tex]\[ y = ax^2 + bx + c \][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants, and [tex]\( a \neq 0 \)[/tex].
Let's examine each given option:
1. [tex]\( y = 2^x + 4 \)[/tex]:
- This function is an exponential function because the variable [tex]\( x \)[/tex] is in the exponent. It does not fit the form of a quadratic function.
2. [tex]\( y = 3x - 2 \)[/tex]:
- This is a linear function because it can be written in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] and [tex]\( b \)[/tex] are constants. Here, there is no [tex]\( x^2 \)[/tex] term, so it is not a quadratic function.
3. [tex]\( x^3 - 10x = 21 \)[/tex]:
- This can be rewritten as [tex]\( x^3 - 10x - 21 = 0 \)[/tex]. It is a cubic equation because the highest power of [tex]\( x \)[/tex] is 3. This is not a quadratic function as it does not fit the form [tex]\( y = ax^2 + bx + c \)[/tex].
4. [tex]\( y = 6 + 5x + x^2 \)[/tex]:
- This equation can be rewritten as [tex]\( y = x^2 + 5x + 6 \)[/tex]. It is in the form [tex]\( y = ax^2 + bx + c \)[/tex] with [tex]\( a = 1 \)[/tex] (which is not zero), [tex]\( b = 5 \)[/tex], and [tex]\( c = 6 \)[/tex]. This correctly represents a quadratic function.
Hence, the function that represents a quadratic function is:
[tex]\[ y = 6 + 5x + x^2 \][/tex]
So the correct answer is option 4.
