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 = 4x²

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.

feet	0	1	2	3	4
inches	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 $y = mx + b$ (slope - intercept form) or $Ax+By = C$ (standard form), where $m$, $b$, $A$, $B$, and $C$ are constants and the highest power of the variable is 1.

Step2: Analyze function 1

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

Step3: Analyze function 2

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

Step4: Analyze function 3

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

Step5: Analyze function 4

For $y = 4x^{2}$, the highest power of $x$ is 2. So it is a non - linear function.

Step6: Analyze graph 5

The graph is a parabola (a U - shaped curve), which is the graph of a quadratic function ($y=ax^{2}+bx + c$, $a
eq0$). So it represents a non - linear function.

Step7: Analyze graph 6

The graph is a straight line. So it represents a linear function.

Step8: Analyze graph 7

The graph is a non - straight curve (it has a bend), so it represents a non - linear function.

Step9: Analyze graph 8

The graph is a horizontal line, which is of the form $y = k$ (a constant function, a special case of a linear function $y=0x + k$). So it represents a linear function.

Step10: Analyze 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 because the rate of change (the ratio of the change in $y$ to the change in $x$) is constant ($\frac{\Delta y}{\Delta x}=12$).

Step11: Analyze table 10

The rate of change of velocity with respect to time is not constant. For example, from $t = 5$ s to $t = 10$ s, the change in velocity is $60000 - 50000=10000$ mph and the change in time is $10 - 5 = 5$ s, so the rate of change is $\frac{10000}{5}=2000$ mph/s. From $t = 10$ s to $t = 15$ s, the change in velocity is $70000 - 60000 = 10000$ mph and the change in time is $15 - 10=5$ s, rate of change is $\frac{10000}{5}=2000$ mph/s, but from $t = 15$ s to $t = 20$ s, the change in velocity is $60000 - 70000=-10000$ mph and the change in time is $20 - 15 = 5$ s, rate of change is $\frac{-10000}{5}=-2000$ mph/s. 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 $y=-2x + 10$ which is in $y = mx + b$ form.
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 a non - straight curve.
8. Linear function. Justification: The graph is a horizontal line (a special case of linear function).
9. Linear function. Justification: The relationship is $y = 12x$ with a constant rate of change.
10. Non - linear function. Justification: The rate of change of velocity with respect to time is not constant.