Question

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.
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) 50000 60000 70000 60000 50000

Explanation:

Step1: Recall linear - function form

A linear function is of 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 $y = 3x$

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

Step3: Analyze $y=-2 + 5x$

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

Step4: Analyze $2x + y=10$

We can rewrite it 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 $y = 4x^{2}$

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 the graph of a quadratic function. 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. So it represents a non - linear function.

Step9: Analyze graph 8

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

Step10: Analyze inches - feet relationship

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 in the form $y=mx + b$ with $m = 12$ and $b = 0$. So it is a linear function.

Step11: Analyze Cassini 2 velocity

The velocity values do not change at a constant rate. If it were linear, the change in velocity with respect to time would be constant. Since the velocity first increases, then decreases, it is a non - linear function.

Answer:

1. Linear function.
2. Linear function.
3. Linear function.
4. Non - linear function.
5. Non - linear function.
6. Linear function.
7. Non - linear function.
8. Linear function.
9. Linear function.
10. Non - linear function.