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Which statement best explains whether the equation [tex] y = \frac{1}{2} x - 2 [/tex] represents a linear or nonlinear function?

A. The equation represents a nonlinear function because it has an independent and a dependent variable, each with an exponent of 1.

B. The equation represents a nonlinear function because its graph contains the points [tex] (-1, -2), (0, 0) [/tex], and [tex] (1, 2) [/tex], which are not on a straight line.

C. The equation represents a linear function because it has an independent and a dependent variable, each with an exponent of 1.

D. The equation represents a linear function because its graph contains the points [tex] (-1, -2), (0, 0) [/tex], and [tex] (1, 2) [/tex], which are on a straight line.



Answer :

GinnyAnswer
To determine whether the equation [tex]\( y = \frac{1}{2}x - 2 \)[/tex] represents a linear or nonlinear function, let's analyze its structure and characteristics.

1. Form of the Equation: The given equation is [tex]\( y = \frac{1}{2}x - 2 \)[/tex]. This resembles the form [tex]\( y = mx + b \)[/tex] where:
- [tex]\( m \)[/tex] is the slope of the line.
- [tex]\( b \)[/tex] is the y-intercept.

2. Exponents of Variables: In the equation [tex]\( y = \frac{1}{2}x - 2 \)[/tex]:
- The exponent of [tex]\( x \)[/tex] is 1.
- The exponent of [tex]\( y \)[/tex] is also implicitly 1 since it is understood as [tex]\( y^1 \)[/tex].

3. Linear Equations: An equation is considered linear if it can be written in the form [tex]\( y = mx + b \)[/tex] and if the exponents of the variables are 1. The absence of exponents greater than 1, absence of product of variables, and absence of variables in the denominator confirm that the variable terms are linear.

4. Graph and Points: Let's briefly verify the given points:
- Substitute [tex]\( x = -1 \)[/tex]:
[tex]\[ y = \frac{1}{2}(-1) - 2 = -\frac{1}{2} - 2 = -\frac{1}{2} - \frac{4}{2} = -\frac{5}{2} \neq -2 \][/tex]
It seems there is an error with the given points because when [tex]\( x = -1 \)[/tex], [tex]\( y \neq -2 \)[/tex].

5. Correct Statement: Despite the erroneous points provided, the underlying nature of the equation [tex]\( y = \frac{1}{2}x - 2 \)[/tex] confirms that it is a linear equation, based on its format and the exponents of the variables.

Hence, the correct statement is:

The equation represents a linear function because it has an independent and a dependent variable, each with an exponent of 1.

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