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Algebra 1

5.9.3 Graph Equations to Solve Problems

Algebra 15.9.3 Graph Equations to Solve Problems

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Activity

1. The thickness t t in millimeters of a folded sheet of paper after it is folded n n times is given by the equation t = ( 0.05 ) · 2 n t = ( 0.05 ) · 2 n .

a. What does the number 0.05 represent in the equation?

b. What does the number 2 represent in the equation?

c. Use graphing technology to graph the equation t = ( 0.05 ) · 2 n t = ( 0.05 ) · 2 n .

d. How many folds does it take before the folded sheet of paper is more than 1 mm thick? How many folds before it is more than 1 cm thick? Explain how you know.

2. The area of a sheet of paper is 93.5 square inches.

a. Find the remaining visible area of the sheet of paper after it is folded in half once, twice, and three times.

b. Write an equation expressing the visible area a a of the sheet of paper in terms of the number of times it has been folded, n n .

c. Use graphing technology to graph the equation.

d. In this context, can n n take negative values? Be prepared to show your reasoning.

e. Can a a take negative values? Be prepared to show your reasoning.

f. Identify the domain of the graph using an inequality.

g. Identify the range of the graph using an inequality.

h. What equation represents the horizontal asymptote?

3. Use the equation to make a prediction of the thickness with 7 folds.

4. The y y -intercept represents the initial thickness of the paper t t before folding (when n n is zero). Calculate the value of the y y -intercept.

5. The asymptote represents the possible limit of thickness of the paper as it is folded. What does this mean?

Self Check

The graph below represents the function that gives the value of a computer in dollars, y , as a function of time, x , measured in years since the time of purchase. 

What is the range for this graph and situation?

Graph of a decreasing exponential function, x y plane, origin O.

  1. y 0
  2. \(0<y 400\)<br="" \leq=""></y>
  3. x 0
  4. y 400

Additional Resources

Domain and Range of Exponential Graphs

The graph below describes the amount of caffeine, c c , in a person’s body t t hours after an initial measurement of 100 mg. The equation of this line is y = 100 ( 1 0.1 ) t y = 100 ( 1 0.1 ) t .

A graph showing caffeine (mg) decreasing over time (hours); caffeine starts at 100 mg and gradually declines as time increases from 0 to 14 hours. A point P is marked on the curve.

What is the smallest value c c can reach?

Since c c is the amount of caffeine, it can get close to 0.

Can t t  ever take on negative values?

No, time will not be negative in this relationship because time is moving forward.

What are the lowest and highest values for t t ?

At the beginning, t = 0 t = 0 , and t t  can go as long as it takes for c c  to get so close to 0 it is insignificant.

The domain (or values of t t ) written as an inequality is t 0 t 0 .

The range (or values of c c ) written as an inequality is 0 < c 100 0 < c 100 .

Try it

Try It: Domain and Range of Exponential Graphs

The dollar value of a car is a function, f f , of the number of years, t t , since the car was purchased. The car was purchased at $12,000. Tell the domain and range. The equation of this line is y = 12 , 000 ( 1 0.25 ) t y = 12 , 000 ( 1 0.25 ) t .

( t t is exponent)

Line graph showing a downward curve; x-axis labeled Years Since Purchase (0 to 6), y-axis labeled Value in Dollars (0 to 14,000). Value starts near $12,000 and declines rapidly over time.

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