Algebra 1
Inverse Variation #1
Name:_____________________________
1. We have studied many different functions this year. Match the graphs below with the term
that describes it best. Put the letter of the description in the third quadrant of the graph.
A. Quadratic Function
B. Absolute Value Function
a.
C. Exponential Growth Function
D. Exponential Decay Function
E. Increasing Linear Function
F. Decreasing Linear Function
b. Now go back and circle the y-intercept of each of the graphs above. Note that every function
we have studied thus far this year has a y-intercept.
Many families take road trips during the summertime. Driving at slower speeds can
sometimes save gasoline and be safer, but driving faster can save time.
2. For example, a 300-mile trip takes 6 hours at 50 miles per hour, but only 5 hours at 60 miles
per hour. How would the driving time change if the average speed were 40 miles per hour?
Would the driving time increase or decrease?
Suppose that your family is planning a 250-mile trip by car to visit relatives in Eastern
Washington. You are deciding whether to drive over I-90 or HWY 2. Your average speed
could vary from as little as 10 miles per hour if there is traffic or snow going over HWY 2
to 70 miles per hour, on a clear day with no traffic over I-90.
3a. Recall the formula: Distance = Rate • Time. How many hours will a 250-mile trip take if you
average… (round to the nearest 10th if necessary.)
…10 miles per hour?
…20 miles per hour?
…40 miles per hour?
…60 miles per hour?
b. Complete the following table using your results. Fill in any missing answers.
Speed (mph)
Time (hours)
5
10
20
25
40
60
70
4a. Graph your results using the graph to the
right. Use the scale = 5 for both the x and
y-axes. Make sure you label your axes
with numbers AND names!
b. What do you notice about the pattern or
shape of the graph. Describe the shape or
pattern using math terminology.
c. Estimate from your graph how fast you
would need to drive, on average, to make
the trip in 5 hours.
d. Estimate from your graph how much time it
will take if you average 30 miles per hour.
e. This graph is similar to what type of function we have studied this year?
Although the shape of the graph is similar to that of an exponential equation decay
function, the graph you have made is different in some important ways from any other
functions we studied this year, let’s look at those now.
5a. First let’s look at the equation for this situation. To the right,
transform the formula D = rt to solve for t. (Get the t by itself.)
b. Rewrite your equation with y = t and x= r and with 250 substituted in for D, because the trip
we are looking at is 250 miles long.
c. Does this equation look like an exponential decay function? What about it is different?
d. Type this equation into your calculator and look at a table of values. Change your table setup
to have the change in x go by 5 mph.
e. What pattern of change do you notice in the table? Where are the changes in y the largest?
Where are they the smallest?
f. Is this pattern the same as an exponential decay function?
g. What is the value of our function when x = 0?
h. What could you say about our road trip situation when x = 0?
i. How could you have looked at the equation and known that the graph would not be defined
at x= 0?
j. Recall from exponential decay functions, did they have a y-intercept or a value at x = 0?
6. Our new type of function, with x in the denominator, are called inverse variation
functions. We have found one similarity between an inverse variation and an
exponential decay function and two differences. List these below:
Similarity
•
Differences
•
•
Let’s explore a second example of inverse variation functions:
7. To earn a little extra money, you take on the job of mowing your neighbor’s, the Johnson’s,
lawn for the summer. Their lawn is a rectangle that is 72 square yards.
a. List a few possible dimensions (base and height) for a lawn with 72 square yards.
b. Write the equation for finding the area of a rectangle using A = area, b = base, and h = height.
c. Transform this equation to get b by itself.
d. Rewrite your equation with y = b and x= h and with 72 substituted in for A, because the area
of our rectangle is a constant 72 square yards.
e. Type this equation into your calculator and look at a table of values. Change your table setup
to have the change in x go by 1.
f. When the height (x) changes from 1 yd to 2 yds, how much did the base length (y) decrease?
g. When the height (x) changes from 5 yds to 6 yds, how much did the base length (y) decrease?
h. Fill in the following blanks:
As the height (x) increases, the base (y) _______________ at a _________________ rate.
So far we have only looked at examples where positive x and y values are reasonable. Let’s
look at the whole graph of an inverse variation.
8a. Enter the most basic form of an inverse variation
!
equation into your calculator: 𝑦 = !
b. Look at this equation in all 4 quadrants (use a window
that goes from -5 to 5 on both the y-and x-axes).
Sketch that graph to the right.
c. The graph is in which quadrants?
d. Why do negative x values have negative y values? (hint:
think about replacing x in the function with a negative number.)
!
!
e. Leaving the parent function in for y1, enter 𝑦! = ! , and 𝑦! = ! . How do these graphs
compare to our parent function?
f. Clear out all equations and type in this inverse
!
variation function: 𝑦 = ! !
g. Sketch that graph here.
h. The graph is in which quadrants?
i. Why do negative x values have positive y values? (hint: again,
look at the equation and what happens mathematically to the x value.)
!
!
!
!
j. Predict which quadrants you think the graph of 𝑦 = ! ! and 𝑦 = ! ! would be in.
k. Check on your calculator to see if you were correct.
l. Predict which quadrants you think the graph of 𝑦 = ! ! and 𝑦 = ! ! would be in.
m. Check on your calculator to see if you were correct.
n. Do you think any of the graphs we looked at have a y-intercept or are do you think they are
all undefined for x = 0?
o. Have each member of your group check a different equation that we have looked at to see if
you can find any inverse variation functions with a y-intercept.
9. When you swim in the ocean, it can seem very cold, but the warmest part of the ocean is near
the surface. The temperatures get colder and colder the deeper you go. The following graph
shows ocean temperatures at various depths.
a. Regina looked at the graph and thought it might be
exponential decay. She did exponential regression and
got the equation y=39.6(0.999)x. Type this model into
the calculator and look on a table to complete the
following ordered pairs.
(500,
) (1000,
) (2000,
)
(Change the table start to these x values to do this
quickly, or make your table go by 500)
b. Robert looked at the graph and thought it might be an
inverse variation. He did regression on the calculator
!"###
and got the equation was 𝑦 = ! . Type this model
into you calculator and look on a table to complete the
following ordered pairs.
(500,
) (1000,
) (2000,
)
c. Compare Regina and Robert’s ordered pairs with the ordered pairs in the graph. Who did the
better job with their model?
10. A large Seattle middle school has grades 6 – 8, as an end of the year celebration the 8th
graders get to go to a water slide park. The PTSA rents out the park for $5000. This $5000
fee has to be shared equally among all of the students who attend.
a. What will this celebration cost per student if…
…only 20 students attend?
…only 100 students attend?
…only 500 students attend?
b. Write an equation that gives cost per student (y) for any number of students attending (x).
c. What is the difference in the price per student when the number of students changes from 10 to 20?
d. What is the difference in the price per student when the number of students changes from 110
to 120?
e. How many students need to attend for the trip to be reasonably priced? (pick what you think
is reasonable) What is that price?