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 remember:
1.
Linear Functions: These are functions where the highest power of the variable (usually \( x \)) 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 is greater than 1, or the equation includes terms that are not linear (e.g., squares, cubes, square roots, etc.).
Let's analyze each equation step by step:
---
Equation 1: \( y = 3x - 1 \)
- This is in the form \( y = mx + b \) with \( m = 3 \) and \( b = -1 \).
-
Conclusion: Linear
Equation 2: \( 2y + x^3 = 9 \)
- 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: \( 5x^2 - 3x + 1 = y \)
- The term \( 5x^2 \) indicates a quadratic term, which is nonlinear.
-
Conclusion: Nonlinear
Equation 5: \( y + 4 = x \)
- Rearrange to \( y = x - 4 \), which 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 \( y = x^2 + \frac{1}{2} \). 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 \( y = 0x + 8 \). It is linear.
-
Conclusion: Linear
Equation 10: \( 5y = -x \)
- Rearrange to \( y = -\frac{1}{5}x \), which 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 \( y = x^3 - 3x \). The term \( x^3 \) indicates a cubic term, which is nonlinear.
-
Conclusion: Nonlinear
Equation 12: \( x^2 - 4x = 6 + y \)
- Rearrange to \( y = x^2 - 4x - 6 \). 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 \( y = x^2 + 5 \). 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
---
Final Answers
1. Linear
2. Nonlinear
3. Nonlinear
4. Nonlinear
5. Linear
6. Linear
7. Nonlinear
8. Nonlinear
9. Linear
10. Linear
11. Nonlinear
12. Nonlinear
13. Linear
14. Nonlinear
15. Linear
Boxed Answer
\[
\boxed{
\begin{array}{ccc}
1. \text{Linear} & 2. \text{Nonlinear} & 3. \text{Nonlinear} \\
4. \text{Nonlinear} & 5. \text{Linear} & 6. \text{Linear} \\
7. \text{Nonlinear} & 8. \text{Nonlinear} & 9. \text{Linear} \\
10. \text{Linear} & 11. \text{Nonlinear} & 12. \text{Nonlinear} \\
13. \text{Linear} & 14. \text{Nonlinear} & 15. \text{Linear} \\
\end{array}
}
\]
Parent Tip: Review the logic above to help your child master the concept of linear functions worksheet with answers.
