Question

in which equation is y a linear function of x?
a. ( y = 4x - 8 )
b. ( y = 4x^2 - 8 )
c. ( y = 4xy - 8 )
d. ( y = \frac{4}{x} - 8 )

Explanation:

Step1: Recall linear function form

A linear function of \( x \) has the form \( y = mx + b \), where \( m \) and \( b \) are constants, and \( x \) is to the first power.

Step2: Analyze Option A

For \( y = 4x - 8 \), it matches \( y = mx + b \) with \( m = 4 \) and \( b = -8 \). \( x \) is to the first power.

Step3: Analyze Option B

For \( y = 4x^2 - 8 \), \( x \) is squared (\( x^2 \)), so it's a quadratic function, not linear.

Step4: Analyze Option C

For \( y = 4xy - 8 \), it has a term \( xy \) (product of \( x \) and \( y \)), so it's not linear (linear functions have only \( x \) to the first power and no products of \( x \) and \( y \)).

Step5: Analyze Option D

For \( y = \frac{4}{x}-8 = 4x^{-1}-8 \), \( x \) is to the power of -1, so it's a rational function, not linear.

Answer:

A. \( y = 4x - 8 \)