Question

classify the following in their correct classification (linear or non - linear) (1 point) linear

• $y = 7 \times 4x$
• $y = (2x + 5)^2$
• $y = 10x^2$
• $y = 5x - 3$

Explanation:

Step1: Recall linear function form

A linear function has the form \( y = mx + b \), where \( m \) and \( b \) are constants, and the highest power of \( x \) is 1. Non - linear functions have \( x \) with a power other than 1 (like \( x^2 \), \( x^3 \), etc.) or are products/operations that result in a non - first - degree term.

Step2: Analyze \( y = 7\times4x \)

Simplify \( y = 7\times4x \), we get \( y = 28x \). This is in the form \( y=mx + b \) with \( m = 28 \) and \( b = 0 \), so it is linear.

Step3: Analyze \( y=(2x + 5)^2 \)

Expand \( (2x + 5)^2=(2x)^2+2\times2x\times5 + 5^2=4x^2+20x + 25 \). The highest power of \( x \) is 2, so it is non - linear.

Step4: Analyze \( y = 10x^2 \)

The highest power of \( x \) is 2, so it is non - linear.

Step5: Analyze \( y = 5x-3 \)

This is in the form \( y=mx + b \) with \( m = 5 \) and \( b=-3 \), so it is linear.

Answer:

The linear functions are \( y = 7\times4x \) (or \( y = 28x \)) and \( y = 5x-3 \). The non - linear functions are \( y=(2x + 5)^2 \) and \( y = 10x^2 \). If we are only asked to identify linear ones from the given, the linear equations are \( y = 7\times4x \) and \( y = 5x - 3 \).