Question

practice
example 1
determine whether each function is a linear function. justify your answer.

1. y = 3x
2. y=-2 + 5x
3. 2x + y = 10
4. y = 4x^2

example 2
determine whether each graph represents a linear or nonlinear function.
5.
6.
7.
8.
example 3

1. measurement the table shows a function modeling the number of inches and feet. can the relationship be modeled by a linear or nonlinear function? explain.

inches	0	1	2	3	4
feet	0	12	24	36	48

1. astronomy the table shows the velocity of cassini 2 space probe as it passes saturn. is the velocity modeled by a linear or nonlinear function? explain.

time (s)	5	10	15	20	25
velocity (mph)	50,000	60,000	70,000	60,000	50,000

Explanation:

Step1: Recall linear - function form

A linear function has the form $y = mx + b$ (where $m$ and $b$ are constants and the highest power of the variable is 1).

Step2: Analyze function 1 ($y = 3x$)

It is in the form $y=mx + b$ with $m = 3$ and $b = 0$. So it is a linear function.

Step3: Analyze function 2 ($y=-2 + 5x$)

It is in the form $y=mx + b$ with $m = 5$ and $b=-2$. So it is a linear function.

Step4: Analyze function 3 ($2x + y=10$)

Rewrite it as $y=-2x + 10$. It is in the form $y=mx + b$ with $m=-2$ and $b = 10$. So it is a linear function.

Step5: Analyze function 4 ($y = 4x^{2}$)

The highest - power of $x$ is 2. So it is not a linear function.

Step6: Analyze graph 5

The graph is a parabola (a U - shaped curve), which is a non - linear function.

Step7: Analyze graph 6

The graph is a straight line, which represents a linear function.

Step8: Analyze graph 7

The graph has a curved shape, so it represents a non - linear function.

Step9: Analyze graph 8

The graph is a horizontal line of the form $y = c$ (where $c$ is a constant), which is a linear function.

Step10: Analyze inches - feet relationship (table 9)

The relationship between inches and feet is $y = 12x$ (where $y$ is the number of inches and $x$ is the number of feet). It is a linear function.

Step11: Analyze Cassini 2 velocity (table 10)

The velocity does not change at a constant rate. For example, from $t = 5$ to $t = 10$, the change in velocity is $60000 - 50000=10000$, and from $t = 10$ to $t = 15$, the change is $70000 - 60000 = 10000$, but from $t = 15$ to $t = 20$, the change is $60000 - 70000=-10000$. So it is a non - linear function.

Answer:

1. Linear function. Justification: It is in the form $y = mx + b$ with $m = 3$ and $b = 0$.
2. Linear function. Justification: It is in the form $y=mx + b$ with $m = 5$ and $b=-2$.
3. Linear function. Justification: Can be rewritten as $y=-2x + 10$.
4. Non - linear function. Justification: The highest power of $x$ is 2.
5. Non - linear function. Justification: The graph is a parabola.
6. Linear function. Justification: The graph is a straight line.
7. Non - linear function. Justification: The graph is curved.
8. Linear function. Justification: It is a horizontal line of the form $y = c$.
9. Linear function. Justification: The relationship is $y = 12x$.
10. Non - linear function. Justification: The velocity does not change at a constant rate.