Question
Which function has the slowest rate of growth as x approaches infinity?(1 point)
Responses
y=−4x2
y is equal to negative 4 x squared
y=2(1.3)x
y is equal to 2 times 1 point 3 to the x th power
y=5(0.75)x
y is equal to 5 times 0 point 7 5 to the x th power
y=10x2
Responses
y=−4x2
y is equal to negative 4 x squared
y=2(1.3)x
y is equal to 2 times 1 point 3 to the x th power
y=5(0.75)x
y is equal to 5 times 0 point 7 5 to the x th power
y=10x2
Answers
GPT-4o mini
To determine which function has the slowest rate of growth as \( x \) approaches infinity, we need to analyze the behavior of each function:
1. \( y = -4x^2 \)
- This function is a quadratic polynomial that approaches negative infinity as \( x \) increases. However, in terms of growth (where we typically consider positive growth rates), it does not grow positively.
2. \( y = 2(1.3)^x \)
- This is an exponential function with a base greater than 1 (1.3). As \( x \) approaches infinity, this function grows exponentially.
3. \( y = 5(0.75)^x \)
- This is also an exponential function, but the base (0.75) is less than 1. As \( x \) goes to infinity, this function approaches 0, meaning it decays rather than grows.
4. \( y = 10x^2 \)
- This is another quadratic polynomial that grows positively as \( x \) approaches infinity, specifically it grows without bound as \( x \) increases.
From this analysis:
- The function \( y = 5(0.75)^x \) decays to 0 as \( x \) approaches infinity.
- The linear and polynomial functions \( y = -4x^2 \) and \( y = 10x^2 \) grow positively (or negatively in the first case).
- The exponential function \( y = 2(1.3)^x \) also grows exponentially as \( x \) increases.
Based on this, the function with the slowest rate of growth (or decay towards zero) as \( x \) approaches infinity is:
**\( y = 5(0.75)^x \)** (it approaches 0)
1. \( y = -4x^2 \)
- This function is a quadratic polynomial that approaches negative infinity as \( x \) increases. However, in terms of growth (where we typically consider positive growth rates), it does not grow positively.
2. \( y = 2(1.3)^x \)
- This is an exponential function with a base greater than 1 (1.3). As \( x \) approaches infinity, this function grows exponentially.
3. \( y = 5(0.75)^x \)
- This is also an exponential function, but the base (0.75) is less than 1. As \( x \) goes to infinity, this function approaches 0, meaning it decays rather than grows.
4. \( y = 10x^2 \)
- This is another quadratic polynomial that grows positively as \( x \) approaches infinity, specifically it grows without bound as \( x \) increases.
From this analysis:
- The function \( y = 5(0.75)^x \) decays to 0 as \( x \) approaches infinity.
- The linear and polynomial functions \( y = -4x^2 \) and \( y = 10x^2 \) grow positively (or negatively in the first case).
- The exponential function \( y = 2(1.3)^x \) also grows exponentially as \( x \) increases.
Based on this, the function with the slowest rate of growth (or decay towards zero) as \( x \) approaches infinity is:
**\( y = 5(0.75)^x \)** (it approaches 0)