Testinar Grade 8 math

Linear vs. Nonlinear Functions

Learn to reason about and solve Grade 8 problems involving Linear vs. Nonlinear Functions.

CH04-04 Functions and Linear Relationships

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Directions Solve each problem. Show your reasoning on the page or on separate paper. Check the answer key only after finishing.

Which table represents a LINEAR function?
1. Yes, the first differences are constant
2. No, the $y$-values are not all even
3. Yes, all $y$-values are positive
4. No, the $x$-values increase
Which table represents a NONLINEAR function?
1. No, $y$ increases
2. Yes, the first differences are not constant
3. Yes, the $y$-values are doubling
4. No, the $x$-values are equally spaced
Which graph represents a LINEAR function?
1. No, the relationship is not a function
2. Yes, the graph is a straight line
3. No, the slope is positive
4. Yes, it passes through the origin
Which graph represents a NONLINEAR function?
1. Yes, the curve is parabolic
2. No, it starts at the origin
3. Yes, it only has positive $y$-values
4. No, the points are integers
Looking at the table below, is the function linear or nonlinear?
1. Linear, because $y$ increases
2. Nonlinear, first differences are not constant
3. Linear, because the table has 4 points
4. Nonlinear, because $y > x$
Which table shows a LINEAR function?
1. Yes; the $y$-values increase by 6 each step
2. No; $x$ increases by 2 each time
3. No; $y$ is greater than $x$
4. No; the equation is $y=3x-1$
The table shows the distance traveled over time. Is this a linear relationship?
1. No, time goes up to 4
2. Yes, the first differences are constant (60 mi)
3. No, the distances double
4. Yes, the ratio of distance to time is not constant
The table below shows costs. Identify whether the relationship is linear.
1. Yes, each item costs $12
2. No, the cost is too high
3. No, costs only go up
4. Yes, because the total cost increases

Teacher and parent reference

Answer key and explanations

A. Yes, the first differences are constant
For a linear function, the differences between consecutive $y$-values are constant. Here: $7-3=4$, $11-7=4$, $15-11=4$. A constant first difference indicates linearity.
B. Yes, the first differences are not constant
The differences between consecutive $y$-values are $2-1=1$, $4-2=2$, $8-4=4$. These are not constant, which indicates the function is nonlinear.
B. Yes, the graph is a straight line
A linear function is represented by a straight line on a coordinate plane. This graph shows a straight line, so the function is linear.
A. Yes, the curve is parabolic
A nonlinear function does not form a straight line. This graph shows a parabola (curved shape), characteristic of a quadratic or nonlinear function.
B. Nonlinear, first differences are not constant
The first differences are $3, 5, 7$, which are not constant. When first differences vary, the function is nonlinear.
A. Yes; the $y$-values increase by 6 each step
The $y$-values are $5, 11, 17, 23$. The differences are $11-5=6$, $17-11=6$, and $23-17=6$. Because the difference is constant for equal steps in $x$, the function is linear.
B. Yes, the first differences are constant (60 mi)
The differences are $120-60=60$, $180-120=60$, $240-180=60$. Constant differences mean this is a linear relationship with a speed of 60 mi/h.
A. Yes, each item costs $12
The first differences are constant: $24-12=12$, $36-24=12$, $48-36=12$. This represents a linear relationship where cost increases by $12 per item.