To determine the form in which the quadratic function [tex]\( y = -2x^2 + 5x - 1 \)[/tex] is written, we need to identify its structure.
Quadratic functions can be represented in three main forms:
1. Standard Form: This is expressed as [tex]\( y = ax^2 + bx + c \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants. The quadratic term ([tex]\( ax^2 \)[/tex]), linear term ([tex]\( bx \)[/tex]), and the constant term ([tex]\( c \)[/tex]) are clearly laid out.
2. Vertex Form: This is given by [tex]\( y = a(x - h)^2 + k \)[/tex], where [tex]\( (h, k) \)[/tex] is the vertex of the parabola. In this form, the quadratic part is expressed as a completed square.
3. Factored Form: This is written as [tex]\( y = a(x - r_1)(x - r_2) \)[/tex], where [tex]\( r_1 \)[/tex] and [tex]\( r_2 \)[/tex] are the roots (or zeroes) of the quadratic function. This form emphasizes the x-intercepts.
Given the function:
[tex]\[ y = -2x^2 + 5x - 1 \][/tex]
We can see that it follows the structure of the Standard Form, [tex]\( y = ax^2 + bx + c \)[/tex]:
- Here, [tex]\( a = -2 \)[/tex]
- [tex]\( b = 5 \)[/tex]
- [tex]\( c = -1 \)[/tex]
Hence, the quadratic function [tex]\( y = -2x^2 + 5x - 1 \)[/tex] is written in the standard form.
So, the correct answer is:
[tex]\[
\text{standard}
\][/tex]
