Question

Determine whether it is linear. Justify your answer.
A) y=x to the power of 3
B) y=x to the power 2 -5
C) y=-x+2
D) y=-2x

Answer 1

To determine whether a given equation represents a linear function, we need to check if it can be expressed in the standard form of a linear equation, which is \(y = mx + b\), where \(m\) and \(b\) are constants, and \(m\) is the slope. A linear function graph is a straight line.

Let's analyze each option:

A) \(y = x^3\)
- This is a cubic function because the highest power of \(x\) is 3.
- **Justification**: It is not linear because it does not have the form \(y = mx + b\); instead, it forms a curve.

B) \(y = x^2 - 5\)
- This is a quadratic function because the highest power of \(x\) is 2.
- **Justification**: It is not linear because it does not have the form \(y = mx + b\); it also forms a parabola when graphed.

C) \(y = -x + 2\)
- This can be arranged in the form \(y = mx + b\) where \(m = -1\) and \(b = 2\).
- **Justification**: It is linear because it is in standard form and its graph is a straight line.

D) \(y = -2x\)
- This can also be written as \(y = -2x + 0\) which is in the form \(y = mx + b\) where \(m = -2\) and \(b = 0\).
- **Justification**: It is linear because it is in standard form and its graph is a straight line.

In summary:
- A) Not linear
- B) Not linear
- C) Linear
- D) Linear