Question
Of the functions below, which eventually will have the greatest rate of change compared to the others? (1 point)
Responses
y = 2x − 3
y = 2x − 3
y = x2 + 8x + 1
y = x2 + 8x + 1
y = 3x + 10
Responses
y = 2x − 3
y = 2x − 3
y = x2 + 8x + 1
y = x2 + 8x + 1
y = 3x + 10
Answers
GPT-4o mini
To determine which of the given functions has the greatest rate of change as \( x \) increases, we need to analyze the derivatives of each function. The derivative of a function gives us the rate of change of that function.
1. **For the linear function \( y = 2x - 3 \)**:
- The derivative is \( y' = 2 \).
2. **For the quadratic function \( y = x^2 + 8x + 1 \)**:
- The derivative is \( y' = 2x + 8 \).
- This derivative is linear and increases as \( x \) increases. Therefore, its rate of change will eventually increase without bound as \( x \) becomes very large.
3. **For the linear function \( y = 3x + 10 \)**:
- The derivative is \( y' = 3 \).
Now, comparing the derivatives:
- The first function has a constant rate of change of 2.
- The second function has a rate of change that starts at 8 when \( x = 0 \) and increases indefinitely as \( x \) becomes larger.
- The third function has a constant rate of change of 3.
As \( x \) increases, the rate of change of the quadratic function \( y = x^2 + 8x + 1 \) (which is \( 2x + 8 \)) will eventually surpass the constant rates of the other two functions.
Hence, the function that eventually has the greatest rate of change compared to the others is:
**y = x^2 + 8x + 1**
1. **For the linear function \( y = 2x - 3 \)**:
- The derivative is \( y' = 2 \).
2. **For the quadratic function \( y = x^2 + 8x + 1 \)**:
- The derivative is \( y' = 2x + 8 \).
- This derivative is linear and increases as \( x \) increases. Therefore, its rate of change will eventually increase without bound as \( x \) becomes very large.
3. **For the linear function \( y = 3x + 10 \)**:
- The derivative is \( y' = 3 \).
Now, comparing the derivatives:
- The first function has a constant rate of change of 2.
- The second function has a rate of change that starts at 8 when \( x = 0 \) and increases indefinitely as \( x \) becomes larger.
- The third function has a constant rate of change of 3.
As \( x \) increases, the rate of change of the quadratic function \( y = x^2 + 8x + 1 \) (which is \( 2x + 8 \)) will eventually surpass the constant rates of the other two functions.
Hence, the function that eventually has the greatest rate of change compared to the others is:
**y = x^2 + 8x + 1**