This worksheet helps students distinguish between linear and nonlinear relationships using a variety of representations like graphs, equations, and data tables.
Math worksheet titled Linear vs Nonlinear Relationships featuring graphs, equations, and tables for students to color based on function type.
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Step-by-step solution for: Linear vs. Nonlinear Quick Color
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Show Answer Key & Explanations
Step-by-step solution for: Linear vs. Nonlinear Quick Color
To solve this problem, we need to sort each item into one of two categories: Linear or Nonlinear.
Here is the simple rule to follow:
* Linear: The graph is a perfectly straight line. In an equation, $x$ usually has no exponent (like $x^1$) and isn't inside other functions like square roots. In a table, the $y$ values change by the same amount every time $x$ increases by 1.
* Nonlinear: The graph is curved, bent, or shaped like a circle. In an equation, you see exponents like $x^2$, $x^3$, absolute value bars $|x|$, or square roots $\sqrt{x}$. In a table, the change in $y$ is not constant.
Let's go through them row by row.
1. S-shaped curve: This is curved, not straight. -> Nonlinear
2. Straight line going down: It is perfectly straight. -> Linear
3. Wavy curve: It bends up and down. -> Nonlinear
4. Hill shape (Parabola): It is curved. -> Nonlinear
5. $y = 2x + 20$: The $x$ has no exponent (it is just $x$). This makes a straight line. -> Linear
6. $y = x^2 - 9$: The $x$ is squared ($x^2$). This makes a U-shape. -> Nonlinear
7. $y = \frac{1}{x} + 1$: The $x$ is in the denominator. This creates a curve with two separate parts. -> Nonlinear
8. $y = x$: This is the simplest straight line. -> Linear
9. Circle: A circle is curved. -> Nonlinear
10. V-shape: This has a sharp corner. Straight lines don't have corners. -> Nonlinear
11. Straight line going up: It is perfectly straight. -> Linear
12. Straight line going down: It is perfectly straight. -> Linear
*Tip: Check how much $y$ changes when $x$ goes up by 1.*
13. Table 1:
* $x$ goes $1 \to 2 \to 3$.
* $y$ goes $5 \to 10 \to 15$.
* The $y$ values go up by 5 each time. Since the change is constant, it is Linear.
14. Table 2:
* $x$ goes $-5 \to -4 \to -3$.
* $y$ goes $2 \to 4 \to 8$.
* First change: $4 - 2 = 2$. Second change: $8 - 4 = 4$.
* The change is not the same. It is Nonlinear.
15. Table 3:
* $x$ goes $-2 \to -1 \to 0$.
* $y$ goes $2 \to 1 \to 0$.
* The $y$ values go down by 1 each time. Constant change means it is Linear.
16. Table 4:
* $x$ goes $0 \to 1 \to 2$.
* $y$ goes $1.1 \to 1.2 \to 1.5$.
* First change: $1.2 - 1.1 = 0.1$. Second change: $1.5 - 1.2 = 0.3$.
* The change is not the same. It is Nonlinear.
17. $y = x^3 + 6$: The $x$ is cubed ($x^3$). -> Nonlinear
18. $y = -|x|$: The absolute value bars create a V-shape. -> Nonlinear
19. $y = \frac{x}{2} - 3$: This is the same as $y = 0.5x - 3$. No exponents. -> Linear
20. $y = 7x - 8$: No exponents on $x$. -> Linear
21. Table 1:
* $x$ goes $-2 \to -1 \to 0$.
* $y$ goes $-2 \to -1 \to 0$.
* Change is always +1. -> Linear
22. Table 2:
* $x$ goes $-2 \to -1 \to 0$.
* $y$ goes $4 \to 1 \to 0$.
* First change: $1 - 4 = -3$. Second change: $0 - 1 = -1$.
* Changes are different. -> Nonlinear
23. Table 3:
* $x$ goes $-1 \to 0 \to 1$.
* $y$ goes $1 \to 0 \to 1$.
* First change: $-1$. Second change: $+1$.
* Changes are different. -> Nonlinear
24. Table 4:
* $x$ goes $0 \to 4 \to 8$. (Note: $x$ jumps by 4).
* $y$ goes $2 \to 10 \to 18$.
* First change: $10 - 2 = 8$. Second change: $18 - 10 = 8$.
* The change is constant. -> Linear
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Final Answer:
Linear Relationships:
* Row 1, Box 2 (Straight line graph)
* Row 2, Box 1 ($y = 2x + 20$)
* Row 2, Box 4 ($y = x$)
* Row 3, Box 3 (Straight line graph)
* Row 3, Box 4 (Straight line graph)
* Row 4, Box 1 (Table: 5, 10, 15)
* Row 4, Box 3 (Table: 2, 1, 0)
* Row 5, Box 3 ($y = \frac{x}{2} - 3$)
* Row 5, Box 4 ($y = 7x - 8$)
* Row 6, Box 1 (Table: -2, -1, 0)
* Row 6, Box 4 (Table: 2, 10, 18)
Nonlinear Relationships:
* Row 1, Box 1 (S-curve graph)
* Row 1, Box 3 (Wavy graph)
* Row 1, Box 4 (Hill graph)
* Row 2, Box 2 ($y = x^2 - 9$)
* Row 2, Box 3 ($y = \frac{1}{x} + 1$)
* Row 3, Box 1 (Circle graph)
* Row 3, Box 2 (V-shape graph)
* Row 4, Box 2 (Table: 2, 4, 8)
* Row 4, Box 4 (Table: 1.1, 1.2, 1.5)
* Row 5, Box 1 ($y = x^3 + 6$)
* Row 5, Box 2 ($y = -|x|$)
* Row 6, Box 2 (Table: 4, 1, 0)
* Row 6, Box 3 (Table: 1, 0, 1)
Here is the simple rule to follow:
* Linear: The graph is a perfectly straight line. In an equation, $x$ usually has no exponent (like $x^1$) and isn't inside other functions like square roots. In a table, the $y$ values change by the same amount every time $x$ increases by 1.
* Nonlinear: The graph is curved, bent, or shaped like a circle. In an equation, you see exponents like $x^2$, $x^3$, absolute value bars $|x|$, or square roots $\sqrt{x}$. In a table, the change in $y$ is not constant.
Let's go through them row by row.
Row 1: Graphs
1. S-shaped curve: This is curved, not straight. -> Nonlinear
2. Straight line going down: It is perfectly straight. -> Linear
3. Wavy curve: It bends up and down. -> Nonlinear
4. Hill shape (Parabola): It is curved. -> Nonlinear
Row 2: Equations
5. $y = 2x + 20$: The $x$ has no exponent (it is just $x$). This makes a straight line. -> Linear
6. $y = x^2 - 9$: The $x$ is squared ($x^2$). This makes a U-shape. -> Nonlinear
7. $y = \frac{1}{x} + 1$: The $x$ is in the denominator. This creates a curve with two separate parts. -> Nonlinear
8. $y = x$: This is the simplest straight line. -> Linear
Row 3: Graphs
9. Circle: A circle is curved. -> Nonlinear
10. V-shape: This has a sharp corner. Straight lines don't have corners. -> Nonlinear
11. Straight line going up: It is perfectly straight. -> Linear
12. Straight line going down: It is perfectly straight. -> Linear
Row 4: Tables
*Tip: Check how much $y$ changes when $x$ goes up by 1.*
13. Table 1:
* $x$ goes $1 \to 2 \to 3$.
* $y$ goes $5 \to 10 \to 15$.
* The $y$ values go up by 5 each time. Since the change is constant, it is Linear.
14. Table 2:
* $x$ goes $-5 \to -4 \to -3$.
* $y$ goes $2 \to 4 \to 8$.
* First change: $4 - 2 = 2$. Second change: $8 - 4 = 4$.
* The change is not the same. It is Nonlinear.
15. Table 3:
* $x$ goes $-2 \to -1 \to 0$.
* $y$ goes $2 \to 1 \to 0$.
* The $y$ values go down by 1 each time. Constant change means it is Linear.
16. Table 4:
* $x$ goes $0 \to 1 \to 2$.
* $y$ goes $1.1 \to 1.2 \to 1.5$.
* First change: $1.2 - 1.1 = 0.1$. Second change: $1.5 - 1.2 = 0.3$.
* The change is not the same. It is Nonlinear.
Row 5: Equations
17. $y = x^3 + 6$: The $x$ is cubed ($x^3$). -> Nonlinear
18. $y = -|x|$: The absolute value bars create a V-shape. -> Nonlinear
19. $y = \frac{x}{2} - 3$: This is the same as $y = 0.5x - 3$. No exponents. -> Linear
20. $y = 7x - 8$: No exponents on $x$. -> Linear
Row 6: Tables
21. Table 1:
* $x$ goes $-2 \to -1 \to 0$.
* $y$ goes $-2 \to -1 \to 0$.
* Change is always +1. -> Linear
22. Table 2:
* $x$ goes $-2 \to -1 \to 0$.
* $y$ goes $4 \to 1 \to 0$.
* First change: $1 - 4 = -3$. Second change: $0 - 1 = -1$.
* Changes are different. -> Nonlinear
23. Table 3:
* $x$ goes $-1 \to 0 \to 1$.
* $y$ goes $1 \to 0 \to 1$.
* First change: $-1$. Second change: $+1$.
* Changes are different. -> Nonlinear
24. Table 4:
* $x$ goes $0 \to 4 \to 8$. (Note: $x$ jumps by 4).
* $y$ goes $2 \to 10 \to 18$.
* First change: $10 - 2 = 8$. Second change: $18 - 10 = 8$.
* The change is constant. -> Linear
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Final Answer:
Linear Relationships:
* Row 1, Box 2 (Straight line graph)
* Row 2, Box 1 ($y = 2x + 20$)
* Row 2, Box 4 ($y = x$)
* Row 3, Box 3 (Straight line graph)
* Row 3, Box 4 (Straight line graph)
* Row 4, Box 1 (Table: 5, 10, 15)
* Row 4, Box 3 (Table: 2, 1, 0)
* Row 5, Box 3 ($y = \frac{x}{2} - 3$)
* Row 5, Box 4 ($y = 7x - 8$)
* Row 6, Box 1 (Table: -2, -1, 0)
* Row 6, Box 4 (Table: 2, 10, 18)
Nonlinear Relationships:
* Row 1, Box 1 (S-curve graph)
* Row 1, Box 3 (Wavy graph)
* Row 1, Box 4 (Hill graph)
* Row 2, Box 2 ($y = x^2 - 9$)
* Row 2, Box 3 ($y = \frac{1}{x} + 1$)
* Row 3, Box 1 (Circle graph)
* Row 3, Box 2 (V-shape graph)
* Row 4, Box 2 (Table: 2, 4, 8)
* Row 4, Box 4 (Table: 1.1, 1.2, 1.5)
* Row 5, Box 1 ($y = x^3 + 6$)
* Row 5, Box 2 ($y = -|x|$)
* Row 6, Box 2 (Table: 4, 1, 0)
* Row 6, Box 3 (Table: 1, 0, 1)
Parent Tip: Review the logic above to help your child master the concept of linear or nonlinear worksheet.
