All Algebra II Resources
Example Questions
Example Question #1 : How To Find Direct Variation
varies directly with the square root of
. If
, then
. What is the value of
if
?
None of these answers are correct.
If varies directly with the square root of
, then for some constant of variation
,
If , then
; therefore, the equation becomes
,
or
.
Divide by 5 to get , making the equation
.
If , then
.
Example Question #1 : Identifying Variable Relationships
If varies directly with
and when
,
due to the effect of a constant, what is the value of
when
?
Since varies directly with
,
where
is a constant.
1. Solve for when
and
.
2. Use your equation to solve for when
.
Example Question #2 : Identifying Variable Relationships
If varies indirectly with
and when
,
due to the effect of a constant, what is the value of
when
?
Since varies indirectly with
,
1. Solve for when
and
.
2. Use the equation you found to solve for when
.
Example Question #1 : Variable Relationships
varies directly with
. If
, what is
if
?
1. Since varies directly with
:
with K being some constant.
2. Solve for K using the x and y values given:
3. Use the equation you solved for to find the value of y:
Example Question #4 : Identifying Variable Relationships
varies inversely with
. If
, then what is
equal to when
?
1. Since varies indirectly with
:
2. Use the given x and y values to determine the value of K:
3. Using the equation along with the value of K, find the value of y when x=5:
Example Question #5 : Identifying Variable Relationships
varies directly with
and when
. What is
when
?
1. Since varies directly with
:
2. Use the values given for x and y to solve for K:
3. Use your new equation with the K you solved for to solve for y when x=27:
Example Question #6 : Identifying Variable Relationships
varies inversely with
. When
. What is the value of
when
?
1. Since y varies indirectly with :
2. Solve for K using the x and y values given:
3. Using the equation you created by solving for K, find y when x=100:
Example Question #1 : Interpolations
Given the two following points, use interpolation to determine the best estimate for the value
,
Using our two known points, we can use interpolation to determine the value at any point between them with the following formula:
Where is our first given point,
is our second given point, and
is the point we want to find. We know our two given points, as well as the x value of our unknown point, so now all we must do is plug in all of our known values and solve for y, our only unknown:
Example Question #2 : Interpolations
The output of a factory in units per day versus the number of employees working is plotted on the graph below, with the following data points collected:
(Workers, Units of output per day):
Assuming a linear relationship, interpolate to find how many units will be made per day if workers are present.
We want to do a linear interpolation since the relationship between workers and units can be assumed to be linear. This means there is a constant slope between the points, so the slope between two known points will be equal to the slope between the point we are trying to find and some known point. This is expressed in the relation:
,
where and
are the points we want to find and
and
are known. We choose the known points to be those that are just to the left and right of the point we are trying to find,
and
.
Plugging these into our interpolation formula and knowing , we can find
, the units output per day.
.
Simplifying and rearranging to solve for :
.
So there are units produced when the number of workers is
.
Example Question #3 : Interpolations
Given the points and
, use linear interpolation to find the value of
when
.
Use the formula for interpolation to determine the value of y:
We will use (30, 51) as our x2 and y2 and (20, 36) as our x1 and y1 and we will solve for y using 26.5 for x.
Certified Tutor
All Algebra II Resources
![Learning Tools by Varsity Tutors](https://vt-vtwa-app-assets.varsitytutors.com/assets/problems/og_image_practice_problems-9cd7cd1b01009043c4576617bc620d0d5f9d58294f59b6d6556fd8365f7440cf.jpg)