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 form

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 \(m\) is the slope and the highest power of \(x\) and \(y\) is 1. A nonlinear function has a variable with a power other than 1 (e.g., \(x^2\), \(x^3\), \(x^4\) etc.) or cannot be written in the linear forms.

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

The equation \(y - 4=-8(x - 1)\) is in point - slope form (\(y - y_1=m(x - x_1)\) with \(y_1 = 4\), \(x_1=1\) and \(m=-8\)). We can also rewrite it as \(y=-8x + 8+4=-8x + 12\), which is in the form \(y=mx + b\) (linear form). So this function is linear.

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

The function \(y = x^4\) has the variable \(x\) raised to the power of 4. Since the power of \(x\) is not 1, this function is nonlinear.

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

Rewrite the equation as \(y=x^2 + 4.5\). The variable \(x\) is raised to the power of 2. Since the power of \(x\) is not 1, this function is nonlinear.

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

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

$$
LATEXBLOCK0
$$

This is in the form \(y = mx + b\) (linear form). So this function 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)