Question
A quadratic function has x-intercepts that are 8 units apart. Which of the following options expresses a quadratic function with that feature? Assume that a is a constant. Select all that apply. A. f(x) = (x + a)(x + a + 8)
B. f(x) = x^ 2– 8x + 16
C. f(x) = x ^2– 16
D. f(x) = x ^2+ 8
B. f(x) = x^ 2– 8x + 16
C. f(x) = x ^2– 16
D. f(x) = x ^2+ 8
Answers
GPT-4o mini
To determine which of the given options represents a quadratic function with x-intercepts that are 8 units apart, we need to analyze the x-intercepts of each function.
A quadratic function can be expressed in the form \( f(x) = a(x - r_1)(x - r_2) \), where \( r_1 \) and \( r_2 \) are the x-intercepts. If the x-intercepts are 8 units apart, then we can say that \( |r_1 - r_2| = 8 \).
Let's analyze each option:
**A. \( f(x) = (x + a)(x + a + 8) \)**
The x-intercepts of this function can be found by setting the function equal to zero.
Setting \( (x + a)(x + a + 8) = 0 \) gives \( x + a = 0 \) or \( x + a + 8 = 0 \).
The intercepts are at \( r_1 = -a \) and \( r_2 = -a - 8 \).
The distance between the intercepts is \(|(-a) - (-a - 8)| = |8| = 8\).
This option works.
**B. \( f(x) = x^2 - 8x + 16 \)**
This can be factored as \( f(x) = (x - 4)(x - 4) \), which indicates a double root at \( x = 4 \).
In this case, there is only one x-intercept, which does not meet the requirement of being 8 units apart.
This option does not work.
**C. \( f(x) = x^2 - 16 \)**
Factoring this gives \( f(x) = (x - 4)(x + 4) \), which has x-intercepts at \( x = 4 \) and \( x = -4 \).
The distance between these intercepts is \(|4 - (-4)| = |4 + 4| = 8\).
This option works.
**D. \( f(x) = x^2 + 8 \)**
This function does not cross the x-axis since the minimum value is positive (it is always \( 8 \) or higher).
Thus, there are no x-intercepts.
This option does not work.
In summary, the options that express a quadratic function with x-intercepts that are 8 units apart are:
- **A. \( f(x) = (x + a)(x + a + 8) \)**
- **C. \( f(x) = x^2 - 16 \)**
A quadratic function can be expressed in the form \( f(x) = a(x - r_1)(x - r_2) \), where \( r_1 \) and \( r_2 \) are the x-intercepts. If the x-intercepts are 8 units apart, then we can say that \( |r_1 - r_2| = 8 \).
Let's analyze each option:
**A. \( f(x) = (x + a)(x + a + 8) \)**
The x-intercepts of this function can be found by setting the function equal to zero.
Setting \( (x + a)(x + a + 8) = 0 \) gives \( x + a = 0 \) or \( x + a + 8 = 0 \).
The intercepts are at \( r_1 = -a \) and \( r_2 = -a - 8 \).
The distance between the intercepts is \(|(-a) - (-a - 8)| = |8| = 8\).
This option works.
**B. \( f(x) = x^2 - 8x + 16 \)**
This can be factored as \( f(x) = (x - 4)(x - 4) \), which indicates a double root at \( x = 4 \).
In this case, there is only one x-intercept, which does not meet the requirement of being 8 units apart.
This option does not work.
**C. \( f(x) = x^2 - 16 \)**
Factoring this gives \( f(x) = (x - 4)(x + 4) \), which has x-intercepts at \( x = 4 \) and \( x = -4 \).
The distance between these intercepts is \(|4 - (-4)| = |4 + 4| = 8\).
This option works.
**D. \( f(x) = x^2 + 8 \)**
This function does not cross the x-axis since the minimum value is positive (it is always \( 8 \) or higher).
Thus, there are no x-intercepts.
This option does not work.
In summary, the options that express a quadratic function with x-intercepts that are 8 units apart are:
- **A. \( f(x) = (x + a)(x + a + 8) \)**
- **C. \( f(x) = x^2 - 16 \)**