1. Use the table below to determine whether y varies directly with x. If it does, find the constant of variation k and write the relationship as y=kx.
Please show your work
X
1
2
3
4
У
-4
-8
-12
-16
To determine whether y varies directly with x, we need to check if the ratio of y-values to x-values is constant.
Let's calculate the ratio of y to x for each pair of values:
For x=1, y=-4: y/x = -4/1 = -4
For x=2, y=-8: y/x = -8/2 = -4
For x=3, y=-12: y/x = -12/3 = -4
For x=4, y=-16: y/x = -16/4 = -4
Since the ratio of y to x is constant at -4 for all pairs of values, we can conclude that y varies directly with x.
Now, let's find the constant of variation (k) by picking one pair of values and dividing y by x:
For x=1, y=-4: k = y/x = -4/1 = -4
Therefore, the relationship between y and x is y = -4x.
2. Find the value of y for a given value of x, if y varies directly with x.
If y = 156 when x = 39, what is y when x = 17?
Please show your work
Since y varies directly with x, we can use the constant of variation to find the value of y for a given value of x.
First, let's find the constant of variation (k) using the given values.
When x = 39, y = 156.
We can use the formula y = kx to find k.
156 = k * 39
Divide both sides of the equation by 39:
156/39 = k
k = 4
Now that we know the value of k, we can find y when x = 17.
Using the formula y = kx:
y = 4 * 17
y = 68
Therefore, when x = 17, y = 68.
3. What is the output of the function f(x) for the given input?
f(x) = 3x-11 when x=8
To find the output of the function f(x) for the given input, we can substitute the value of x into the function and evaluate it.
Let's substitute x = 8 into the function f(x) = 3x - 11:
f(8) = 3(8) - 11
f(8) = 24 - 11
f(8) = 13
Therefore, the output of the function f(x) for the input x = 8 is 13.
To determine if y varies directly with x, let's check if the ratio of y to x is constant.
We can calculate the ratios for each pair of corresponding values of x and y:
For the first pair (1, -4): -4/1 = -4
For the second pair (2, -8): -8/2 = -4
For the third pair (3, -12): -12/3 = -4
For the fourth pair (4, -16): -16/4 = -4
Since the ratio is constant for all pairs, we can conclude that y varies directly with x.
To find the constant of variation (k), we can use any of the ratios calculated. For simplicity, we'll use the first pair:
k = y/x = -4/1 = -4
Therefore, the relationship between y and x can be written as y = -4x.
To determine whether y varies directly with x, we need to check if the ratio between y and x values remains constant throughout the table.
Let's calculate the ratio y/x for each pair of values:
For x=1, y= -4: y/x = -4/1 = -4
For x=2, y= -8: y/x = -8/2 = -4
For x=3, y= -12: y/x = -12/3 = -4
For x=4, y= -16: y/x = -16/4 = -4
We can see that the ratio y/x is -4 for all values of x. This indicates that y varies directly with x.
To find the constant of variation k, we can choose any pair of values from the table and divide them. Let's choose x=1 and y=-4:
k = y/x = -4/1 = -4
Therefore, the relationship between y and x is y = -4x.