Question

the table shows values of the expressions $10x^2$ and $2^x$.

1. describe how the values of each expression change as $x$ increases.
2. predict which expression will have a greater value when:

a. $x$ is 8
b. $x$ is 10
c. $x$ is 12

1. find the value of each expression when $x$ is 8, 10, and 12.
2. make an observation about how the values of the two expressions change as $x$ becomes greater and greater.

(the table has columns $x$, $10x^2$, $2^x$ with rows showing $x$ values 1,2,3,4,8,10,12 and corresponding $10x^2$ and $2^x$ values: for $x=1$, $10x^2=10$, $2^x=2$; $x=2$, $10x^2=40$, $2^x=4$; $x=3$, $10x^2=90$, $2^x=8$; $x=4$, $10x^2=160$, $2^x=16$; $x=8,10,12$ have empty $10x^2$ and $2^x$ cells initially.)

Explanation:

Part 1: Describe how the values of each expression change as \( x \) increases

For \( 10x^2 \): As \( x \) increases, \( x^2 \) increases quadratically, so \( 10x^2 \) increases quadratically (the rate of increase accelerates, but it is a polynomial growth).
For \( 2^x \): As \( x \) increases, \( 2^x \) increases exponentially (the rate of increase accelerates much faster than polynomial growth).

Part 2: Predict which expression is greater

a. \( x = 8 \)

• Calculate \( 10x^2 \): \( 10(8)^2 = 10(64) = 640 \)
• Calculate \( 2^x \): \( 2^8 = 256 \)

Since \( 640 > 256 \), \( 10x^2 \) is greater.

b. \( x = 10 \)

• Calculate \( 10x^2 \): \( 10(10)^2 = 10(100) = 1000 \)
• Calculate \( 2^x \): \( 2^{10} = 1024 \)

Since \( 1024 > 1000 \), \( 2^x \) is greater.

c. \( x = 12 \)

• Calculate \( 10x^2 \): \( 10(12)^2 = 10(144) = 1440 \)
• Calculate \( 2^x \): \( 2^{12} = 4096 \)

Since \( 4096 > 1440 \), \( 2^x \) is greater.

Part 3: Find the value of each expression

\( x = 8 \)

• \( 10x^2 \): \( 10(8)^2 = 10 \times 64 = 640 \)
• \( 2^x \): \( 2^8 = 256 \)

\( x = 10 \)

• \( 10x^2 \): \( 10(10)^2 = 10 \times 100 = 1000 \)
• \( 2^x \): \( 2^{10} = 1024 \)

\( x = 12 \)

• \( 10x^2 \): \( 10(12)^2 = 10 \times 144 = 1440 \)
• \( 2^x \): \( 2^{12} = 4096 \)

Part 4: Observation as \( x \) becomes very large

For small \( x \), \( 10x^2 \) (polynomial) may be greater, but as \( x \) increases, \( 2^x \) (exponential) grows much faster than \( 10x^2 \). Eventually, exponential growth outpaces polynomial growth, so \( 2^x \) will be much larger than \( 10x^2 \) for sufficiently large \( x \).

Final Answers (Summarized)

1. \( 10x^2 \): Quadratic growth; \( 2^x \): Exponential growth.
2. a. \( 10x^2 \); b. \( 2^x \); c. \( 2^x \)
3. \( x=8 \): \( 10x^2=640 \), \( 2^x=256 \); \( x=10 \): \( 10x^2=1000 \), \( 2^x=1024 \); \( x=12 \): \( 10x^2=1440 \), \( 2^x=4096 \)
4. Exponential (\( 2^x \)) outpaces polynomial (\( 10x^2 \)) for large \( x \).

Answer:

Part 1: Describe how the values of each expression change as \( x \) increases

For \( 10x^2 \): As \( x \) increases, \( x^2 \) increases quadratically, so \( 10x^2 \) increases quadratically (the rate of increase accelerates, but it is a polynomial growth).
For \( 2^x \): As \( x \) increases, \( 2^x \) increases exponentially (the rate of increase accelerates much faster than polynomial growth).

Part 2: Predict which expression is greater

a. \( x = 8 \)

• Calculate \( 10x^2 \): \( 10(8)^2 = 10(64) = 640 \)
• Calculate \( 2^x \): \( 2^8 = 256 \)

Since \( 640 > 256 \), \( 10x^2 \) is greater.

b. \( x = 10 \)

• Calculate \( 10x^2 \): \( 10(10)^2 = 10(100) = 1000 \)
• Calculate \( 2^x \): \( 2^{10} = 1024 \)

Since \( 1024 > 1000 \), \( 2^x \) is greater.

c. \( x = 12 \)

• Calculate \( 10x^2 \): \( 10(12)^2 = 10(144) = 1440 \)
• Calculate \( 2^x \): \( 2^{12} = 4096 \)

Since \( 4096 > 1440 \), \( 2^x \) is greater.

Part 3: Find the value of each expression

\( x = 8 \)

• \( 10x^2 \): \( 10(8)^2 = 10 \times 64 = 640 \)
• \( 2^x \): \( 2^8 = 256 \)

\( x = 10 \)

• \( 10x^2 \): \( 10(10)^2 = 10 \times 100 = 1000 \)
• \( 2^x \): \( 2^{10} = 1024 \)

\( x = 12 \)

• \( 10x^2 \): \( 10(12)^2 = 10 \times 144 = 1440 \)
• \( 2^x \): \( 2^{12} = 4096 \)

Part 4: Observation as \( x \) becomes very large

For small \( x \), \( 10x^2 \) (polynomial) may be greater, but as \( x \) increases, \( 2^x \) (exponential) grows much faster than \( 10x^2 \). Eventually, exponential growth outpaces polynomial growth, so \( 2^x \) will be much larger than \( 10x^2 \) for sufficiently large \( x \).

Final Answers (Summarized)

1. \( 10x^2 \): Quadratic growth; \( 2^x \): Exponential growth.
2. a. \( 10x^2 \); b. \( 2^x \); c. \( 2^x \)
3. \( x=8 \): \( 10x^2=640 \), \( 2^x=256 \); \( x=10 \): \( 10x^2=1000 \), \( 2^x=1024 \); \( x=12 \): \( 10x^2=1440 \), \( 2^x=4096 \)
4. Exponential (\( 2^x \)) outpaces polynomial (\( 10x^2 \)) for large \( x \).