Comparing Quadratic and Exponential Functions

ID: dorof-sotah
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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2
Grade: 8-9
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29 questions

Comparing Quadratic and Exponential Functions

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Problem 1

The table shows values of the expressions 10x210x^2 and 2x2^x.

1) Describe how the values of each expression change as xx increases.

2) Predict which expression will have a greater value when xx is 8.

3) Predict which expression will have a greater value when xx is 10.

4) Predict which expression will have a greater value when xx is 12.

5) Find the value of 10x210x^2 when xx is 8.

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6) Find the value of 10x210x^2 when xx is 10.

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7) Find the value of 10x210x^2 when xx is 12.

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8) Find the value of 2x2^x when xx is 8.

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9) Find the value of 2x2^x when xx is 10.

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10) Find the value of 2x2^x when xx is 12.

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11) Make an observation about how the values of the two expressions change as xx becomes greater and greater.

Problem 2

Function ff is defined by f(x)=1.5xf(x) = 1.5^x. Function gg is defined by g(x)=500x2+345xg(x) = 500x^2 + 345x.

12) Which function is quadratic?

13) Which one is exponential?

14) The values of which function will eventually be greater for larger and larger values of xx?

Problem 3

15) Create a table of values to show that the exponential expression 3(2)x3(2)^x eventually overtakes the quadratic expression 3x2+2x3x^2 + 2x.

Problem 4

16) The table shows the values of 4x4^x and 100x2100x^2 for some values of xx.

x4x100x21410021640036490042561600510242500\begin{array}{|c|c|c|} \hline \\[-1em] x & 4^x & 100x^2 \\[-1em] \\ \hline \\[-1em] 1 & 4 & 100 \\[-1em] \\ \hline \\[-1em] 2 & 16 & 400 \\[-1em] \\ \hline \\[-1em] 3 & 64 & 900 \\[-1em] \\ \hline \\[-1em] 4 & 256 & 1600 \\[-1em] \\ \hline \\[-1em] 5 & 1024 & 2500 \\[-1em] \\ \hline \end{array}

Use the patterns in the table to explain why eventually the values of the exponential expression 4x4^x will overtake the values of the quadratic expression 100x2100x^2.

Problem 5

Here is a pattern of shapes. The area of each small square is 1 sq cm.

A template for answering this question. Ask your instructor for an alternative.

17) What is the area of the shape in Step 10?

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18) What is the area of the shape in Step nn?

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19) Explain how you see the pattern growing.

Problem 6

A bicycle costs $240 and it loses 35\frac{3}{5} of its value each year.

20) Write an expression for the value of the bicycle, in dollars, after 1 year.

21) Write an expression for the value of the bicycle, in dollars, after 2 years.

22) Write an expression for the value of the bicycle, in dollars, after 3 years.

23) When will the bike be worth less than $1?

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24) Will the value of the bike ever be 0?

True or false? Write below.

25) Explain your reasoning.

Problem 7

A farmer plants wheat and corn. It costs about $150 per acre to plant wheat and about $350 per acre to plant corn. The farmer plans to spend no more than $250,000 planting wheat and corn. The total area of corn and wheat that the farmer plans to plant is less than 1200 acres.

This graph represents the inequality, 150w+350c250,000150w + 350c \leq 250,000, which describes the cost constraint in this situation. Let ww represent the number of acres of wheat and cc represent the number of acres of corn.

A template for answering this question. Ask your instructor for an alternative.

26) The inequality, x+y<1,200x + y < 1,200 represents the total area constraint in this situation. On the same coordinate plane, graph the solution to this inequality.

A template for answering this question. Ask your instructor for an alternative.

27) Use the graphs to find at least two possible combinations of the number of acres of wheat and the number of acres of corn that the farmer could plant.

28) The combination of 400 acres of wheat and 700 acres of corn meets one constraint in the situation but not the other constraint. Which constraint does this meet?

29) Explain your reasoning.