Question

find the error dan claims that because ( y = x + 1 ) and ( y = 3x + 2 ) are both linear functions, the function ( y = (x + 1)(3x + 2) ) must also be linear. is he correct? explain your reasoning. you can choose points on the graph of the function ( y = (x + 1)(3x + 2) ) and show on a coordinate plane that they select choice on a single line.

Explanation:

Step1: Analyze the functions

The function \( y=(x + 1)(3x + 2) \) is a product of two linear functions. Let's expand it: \( y=(x + 1)(3x + 2)=3x^{2}+2x+3x + 2=3x^{2}+5x + 2 \). This is a quadratic function (degree 2), not linear.

Step2: Check linearity

A linear function has the form \( y = mx + b \) (degree 1). The expanded form of \( y=(x + 1)(3x + 2) \) is quadratic (\( 3x^{2}+5x + 2 \)), so it's not linear. Dan's mistake is assuming the product of two linear functions is linear.

Step3: Test points (optional but to confirm)

Take \( x = 0 \): For \( y=x + 1 \), \( y=1 \); for \( y=3x + 2 \), \( y=2 \); for \( y=(x + 1)(3x + 2) \), \( y=(0 + 1)(0 + 2)=2 \). Take \( x = 1 \): For \( y=x + 1 \), \( y=2 \); for \( y=3x + 2 \), \( y=5 \); for \( y=(x + 1)(3x + 2) \), \( y=(1 + 1)(3 + 2)=10 \). The points \((0,1)\), \((0,2)\), \((0,2)\) and \((1,2)\), \((1,5)\), \((1,10)\) don't lie on a single line, confirming it's not linear.

Answer:

Dan is incorrect. The function \( y=(x + 1)(3x + 2) \) expands to \( y = 3x^{2}+5x + 2 \), a quadratic (not linear) function. The product of two linear functions is a quadratic function, so its graph is a parabola, not a line.