Question

Which of the following tables represents a linear function?


x 2 2 2 2 2
y −3 −2 −1 0 1

x −2 −1 0 2 4
y −4 −2 −1 0 1

x −6 −2 0 1 3
y −7 negative one third −5 negative four thirds 1

x −3 −1 1 3 5
y −7 negative nine halves −2 one half 3

Answer 1

Find the slope or you can just pair them into coordinates and see if they become linear

Answer 2

Let's go through each table and see if it represents a linear function.

For a table to represent a linear function, there must be a constant rate of change between any two points. We can check this by looking at the differences in $y$ values when we move from one $x$ value to the next. If the differences are constant, then the function is linear.

Let's start with the first table.

x | 2 | 2 | 2 | 2 | 2
--|---|---|---|---|--
y |-3 |-2 |-1 |0 |1

When we move from $x=2$ to $x=2$ (the same $x$ value), the $y$ value changes from $-3$ to $-2$, so the rate of change is $1$. However, when we move from $x=2$ to $x=2$ again, the $y$ value changes from $-2$ to $-1$, so the rate of change is $1$ again. This means that the function is not linear, since the rate of change is not constant.

Let's move on to the second table.

x |-2 |-1 |0 |2 |4
--|---|---|---|---|--
y |-4 |-2 |-1 |0 |1

When we move from $x=-2$ to $x=-1$, the $y$ value changes from $-4$ to $-2$, so the rate of change is $2$. When we move from $x=-1$ to $x=0$, the $y$ value changes from $-2$ to $-1$, so the rate of change is $1$. When we move from $x=0$ to $x=2$, the $y$ value changes from $-1$ to $0$, so the rate of change is $1/2$. When we move from $x=2$ to $x=4$, the $y$ value changes from $0$ to $1$, so the rate of change is $1/2$ again. Since the rate of change is not constant, the function is not linear.

Let's move on to the third table.

x |-6 |-2 |0 |1 |3
--|---|---|---|---|--
y |-7 |-1/3 |-5 |-4/3 |1

When we move from $x=-6$ to $x=-2$, the $y$ value changes from $-7$ to $-1/3$, so the rate of change is $22/3$. When we move from $x=-2$ to $x=0$, the $y$ value changes from $-1/3$ to $-5$, so the rate of change is $-5/2$. When we move from $x=0$ to $x=1$, the $y$ value changes from $-5$ to $-4/3$, so the rate of change is $7/3$. When we move from $x=1$ to $x=3$, the $y$ value changes from $-4/3$ to $1$, so the rate of change is $13/3$. Since the rate of change is not constant, the function is not linear.

Finally, let's look at the fourth table.

x |-3 |-1 |1 |3 |5
--|---|---|---|---|--
y |-7 |-9/2 |-2 |1/2 |3

When we move from $x=-3$ to $x=-1$, the $y$ value changes from $-7$ to $-9/2$, so the rate of change is $5/2$. When we move from $x=-1$ to $x=1$, the $y$ value changes from $-9/2$ to $-2$, so the rate of change is $5/2$ again. When we move from $x=1$ to $x=3$, the $y$ value changes from $-2$ to $1/2$, so the rate of change is $7/4$. Since the rate of change is not constant, the function is not linear.

Therefore, none of the tables represents a linear function.

Answer 3

let the equation of linear function be y=ax+b
1.it is not a function as x=2 gives different values of y
2.x=0,y=-1
-1=a*0+b
b=-1
y=ax-1
x=2,y=0
0=2a-1
2a=1
a=1/2
y=1/2 x-1
x=4,y=1
1=1/2*4-1
1=2-1
1=1 (true)
x=-2,y=-4
-4=1/2*(-2)-1
-4=-1-1
-4=-1-1
-4=-2
false
it is not a linear function.
3.
y=ax+b
x=0,y=-5
-5=a*0+b
b=-5
y=ax-5
x=-6
y=-7
-7=-6a-5
6a=-5+7
6a=2
a=2/6=1/3
y=1/3 x-5
x=3,y=1
1=1/3*3-5
1=1-5
1=-4 false
not a linear function.
4.
x=-3
y=-7
-7=-3a+b .... (-1)
x=5,y=3
3=5a+b ....(2)
subtract(1) from (2)
10=8a
a=10/8=5/4
from (1)
-7=-3(5/4)+b
b=-7+15/4=(-28+15)/4=-13/4
y=5/4 x-13/4
x=3,y=1/2
1/2=5/4(3)-13/4
1/2=15/4-13/4=2/4=1/2
true
x=-1,=-9/2
-9/2=(5/4)*(-1)-13/4=-5/4-13/4=-18/4=-9/2
true
x=1,y=-2
-2=5/4(1)-13/4=-8/4=-2
true
Hence fourth is alinear function.

Answer 4

Yes, your analysis is correct. The fourth equation is a linear function, while the first three are not linear functions. You have correctly found the slope and y-intercept of the fourth equation, and have tested it with three different points to confirm that it is a linear function. Well done!