Question

select linear or nonlinear to correctly classify each function.
function\tlinear\tnonlinear
y - 4 = -8(x - 1)\t○\t○
y = x⁴\t○\t○
y - x² = 4.5\t○\t○
3x + 5y = 15\t○\t○

Explanation:

Step1: Recall Linear Function Definition

A linear function can be written in the form \( y = mx + b \) (slope - intercept form) or \( y - y_1=m(x - x_1) \) (point - slope form), where \( m \) and \( b \) are constants, and the highest power of \( x \) is 1. A nonlinear function has a highest power of \( x \) greater than 1 or does not follow the linear form.

Step2: Analyze \( y - 4=-8(x - 1) \)

This is in point - slope form \( y - y_1=m(x - x_1) \), where \( m=-8 \), \( x_1 = 1 \), and \( y_1 = 4 \). If we simplify it:

$$
LATEXBLOCK0
$$

The highest power of \( x \) is 1, so it is linear.

Step3: Analyze \( y=x^{4} \)

The highest power of \( x \) is 4, which is greater than 1. So it does not follow the linear function form. Thus, it is nonlinear.

Step4: Analyze \( y - x^{2}=4.5 \)

Rewrite it as \( y=x^{2}+4.5 \). The highest power of \( x \) is 2, which is greater than 1. So it is nonlinear.

Step5: Analyze \( 3x + 5y=15 \)

We can rewrite it in slope - intercept form. Solve for \( y \):

$$
LATEXBLOCK1
$$

The highest power of \( x \) is 1, so it is linear.

Answer:

• For \( y - 4=-8(x - 1) \): Linear (select the circle under "Linear")
• For \( y = x^{4} \): Nonlinear (select the circle under "Nonlinear")
• For \( y - x^{2}=4.5 \): Nonlinear (select the circle under "Nonlinear")
• For \( 3x + 5y = 15 \): Linear (select the circle under "Linear")