Question
the graph of f(x)=ax^2 opens upwards and is narrower than the graph of g(x)=x^2. Which of the following could be a value of a? a.-0.5 b.6 c.0.25 d.-2
Answers
GPT-4o mini
To determine which values of \( a \) make the graph of \( f(x) = ax^2 \) open upwards and be narrower than the graph of \( g(x) = x^2 \), we need to consider the properties of parabolas defined by their leading coefficient.
1. **Opens Upwards**: A parabola opens upwards if its leading coefficient (in this case, \( a \)) is positive. Therefore, \( a \) must be greater than 0.
2. **Narrower than \( g(x) = x^2 \)**: The graph of \( f(x) \) is narrower than that of \( g(x) \) if the absolute value of \( a \) is greater than 1. This means \( a \) must be greater than 1 to be narrower than \( g(x) \).
Now, let's evaluate the options given:
a. \(-0.5\) - This is negative, so the parabola opens downwards. **Not acceptable.**
b. \(6\) - This is positive and greater than 1, meaning the parabola opens upwards and is narrower than \( g(x) \). **Acceptable.**
c. \(0.25\) - This is positive, but it is less than 1. So, the parabola would be wider than \( g(x) \). **Not acceptable.**
d. \(-2\) - This is negative, so the parabola opens downwards. **Not acceptable.**
Thus, the only value that could be a valid **a** is:
**b. 6**
1. **Opens Upwards**: A parabola opens upwards if its leading coefficient (in this case, \( a \)) is positive. Therefore, \( a \) must be greater than 0.
2. **Narrower than \( g(x) = x^2 \)**: The graph of \( f(x) \) is narrower than that of \( g(x) \) if the absolute value of \( a \) is greater than 1. This means \( a \) must be greater than 1 to be narrower than \( g(x) \).
Now, let's evaluate the options given:
a. \(-0.5\) - This is negative, so the parabola opens downwards. **Not acceptable.**
b. \(6\) - This is positive and greater than 1, meaning the parabola opens upwards and is narrower than \( g(x) \). **Acceptable.**
c. \(0.25\) - This is positive, but it is less than 1. So, the parabola would be wider than \( g(x) \). **Not acceptable.**
d. \(-2\) - This is negative, so the parabola opens downwards. **Not acceptable.**
Thus, the only value that could be a valid **a** is:
**b. 6**