Question

4 felipe wants to figure out if the equation $y = x(x + 2)$ represents a linear function.
he finds two (x, y) pairs and plots them.

x	y
2	8

felipe says he can draw a line through these two points, so the equation represents a linear function.
a. explain why felipes reasoning is incorrect.
b. does the equation represent a linear function? explain your reasoning.

Explanation:

Step1: Analyze linear function definition

A linear function has the form $y=mx+b$, where the highest power of $x$ is 1.

Step2: Expand the given equation

$$y = x(x + 2) = x^2 + 2x$$

Step3: Address part (a) reasoning

Two points only define a line, but we need to check if all points from the equation lie on that line. The equation has $x^2$, so most points will not fit the line through the two given points.

Step4: Address part (b) function type

The expanded equation $y=x^2+2x$ has an $x^2$ term (highest power of $x$ is 2), which violates the linear function requirement.

Answer:

a. Felipe's reasoning is incorrect because two points only define a straight line, but the equation produces points that do not all lie on this line. A linear function requires the highest power of $x$ to be 1, which this equation does not satisfy, so most points from the equation will not fit the line through $(-3,3)$ and $(2,8)$.
b. No, the equation does not represent a linear function. When expanded, it becomes $y=x^2+2x$, which is a quadratic function (has an $x^2$ term, highest power of $x$ is 2), not a linear function which must have the form $y=mx+b$ with the highest power of $x$ equal to 1.