Identify Linear And Nonlinear Functions From Equations Worksheet - Free Printable

To determine whether each equation represents a linear or nonlinear function, we need to analyze the form of the equation. Here are the key points to consider:

1. Linear Functions: These are functions where the highest power of the variable \( x \) is 1. The general form is:
\[
y = mx + b
\]
where \( m \) and \( b \) are constants.

2. Nonlinear Functions: These are functions where the highest power of the variable \( x \) is greater than 1, or the equation involves other nonlinear terms such as square roots, exponents, etc.

Let's analyze each equation step by step:

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Equation 1: \( y = 3x - 1 \)

- This is in the form \( y = mx + b \) with \( m = 3 \) and \( b = -1 \).
- Conclusion: Linear

Equation 2: \( 2y + x^3 = 9 \)

- Rearrange to solve for \( y \):
\[
2y = 9 - x^3 \implies y = \frac{9 - x^3}{2}
\]
- The term \( x^3 \) indicates a cubic term, which is nonlinear.
- Conclusion: Nonlinear

Equation 3: \( y = x^2 - 6 \)

- The term \( x^2 \) indicates a quadratic term, which is nonlinear.
- Conclusion: Nonlinear

Equation 4: \( 5x^2 - 3x + 1 = y \)

- This can be rewritten as:
\[
y = 5x^2 - 3x + 1
\]
- The term \( 5x^2 \) indicates a quadratic term, which is nonlinear.
- Conclusion: Nonlinear

Equation 5: \( y + 4 = x \)

- Rearrange to solve for \( y \):
\[
y = x - 4
\]
- This is in the form \( y = mx + b \) with \( m = 1 \) and \( b = -4 \).
- Conclusion: Linear

Equation 6: \( y = \frac{2}{3}x + 5 \)

- This is in the form \( y = mx + b \) with \( m = \frac{2}{3} \) and \( b = 5 \).
- Conclusion: Linear

Equation 7: \( y - \frac{1}{2} = x^2 \)

- Rearrange to solve for \( y \):
\[
y = x^2 + \frac{1}{2}
\]
- The term \( x^2 \) indicates a quadratic term, which is nonlinear.
- Conclusion: Nonlinear

Equation 8: \( y = x^2 \)

- The term \( x^2 \) indicates a quadratic term, which is nonlinear.
- Conclusion: Nonlinear

Equation 9: \( y = 8 \)

- This is a constant function, which can be written as:
\[
y = 0x + 8
\]
- This is a special case of a linear function (horizontal line).
- Conclusion: Linear

Equation 10: \( 5y = -x \)

- Rearrange to solve for \( y \):
\[
y = -\frac{1}{5}x
\]
- This is in the form \( y = mx + b \) with \( m = -\frac{1}{5} \) and \( b = 0 \).
- Conclusion: Linear

Equation 11: \( -y + x^3 = 3x \)

- Rearrange to solve for \( y \):
\[
-y = 3x - x^3 \implies y = -3x + x^3
\]
- The term \( x^3 \) indicates a cubic term, which is nonlinear.
- Conclusion: Nonlinear

Equation 12: \( x^2 - 4x = 6 + y \)

- Rearrange to solve for \( y \):
\[
y = x^2 - 4x - 6
\]
- The term \( x^2 \) indicates a quadratic term, which is nonlinear.
- Conclusion: Nonlinear

Equation 13: \( y = -10x \)

- This is in the form \( y = mx + b \) with \( m = -10 \) and \( b = 0 \).
- Conclusion: Linear

Equation 14: \( x^2 + 5 = y \)

- Rearrange to solve for \( y \):
\[
y = x^2 + 5
\]
- The term \( x^2 \) indicates a quadratic term, which is nonlinear.
- Conclusion: Nonlinear

Equation 15: \( y = \frac{1}{9}x - 4 \)

- This is in the form \( y = mx + b \) with \( m = \frac{1}{9} \) and \( b = -4 \).
- Conclusion: Linear

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Final Answers:

\[
\begin{array}{|c|c|}
\hline
\text{Equation} & \text{Type} \\
\hline
1. y = 3x - 1 & \text{Linear} \\
2. 2y + x^3 = 9 & \text{Nonlinear} \\
3. y = x^2 - 6 & \text{Nonlinear} \\
4. 5x^2 - 3x + 1 = y & \text{Nonlinear} \\
5. y + 4 = x & \text{Linear} \\
6. y = \frac{2}{3}x + 5 & \text{Linear} \\
7. y - \frac{1}{2} = x^2 & \text{Nonlinear} \\
8. y = x^2 & \text{Nonlinear} \\
9. y = 8 & \text{Linear} \\
10. 5y = -x & \text{Linear} \\
11. -y + x^3 = 3x & \text{Nonlinear} \\
12. x^2 - 4x = 6 + y & \text{Nonlinear} \\
13. y = -10x & \text{Linear} \\
14. x^2 + 5 = y & \text{Nonlinear} \\
15. y = \frac{1}{9}x - 4 & \text{Linear} \\
\hline
\end{array}
\]

\boxed{
\begin{array}{|c|c|}
\hline
\text{Equation} & \text{Type} \\
\hline
1. & \text{Linear} \\
2. & \text{Nonlinear} \\
3. & \text{Nonlinear} \\
4. & \text{Nonlinear} \\
5. & \text{Linear} \\
6. & \text{Linear} \\
7. & \text{Nonlinear} \\
8. & \text{Nonlinear} \\
9. & \text{Linear} \\
10. & \text{Linear} \\
11. & \text{Nonlinear} \\
12. & \text{Nonlinear} \\
13. & \text{Linear} \\
14. & \text{Nonlinear} \\
15. & \text{Linear} \\
\hline
\end{array}
}
\)