Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Discover precise answers to your questions from a wide range of experts on our user-friendly Q&A platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
To determine the correct function that represents a parabola with zeros at [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex], and a [tex]\( y \)[/tex]-intercept at [tex]\( (0, -16) \)[/tex], we need to check each of the given functions against the specified conditions.
Condition 1: Zeros at [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex]
A parabola with zeros at [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex] can be written in the factored form as:
[tex]\[ f(x) = a(x + 2)(x - 4) \][/tex]
Condition 2: [tex]\( y \)[/tex]-intercept at [tex]\( (0, -16) \)[/tex]
This means that when [tex]\( x = 0 \)[/tex], [tex]\( f(x) = -16 \)[/tex]. Using this information, we can test each given function.
### Option A: [tex]\( f(x) = 2x^2 + 4x - 16 \)[/tex]
1. Check the zeros:
[tex]\[ f(-2) = 2(-2)^2 + 4(-2) - 16 = 8 - 8 - 16 = -16 \quad (\text{Not } 0) \][/tex]
[tex]\[ f(4) = 2(4)^2 + 4(4) - 16 = 32 + 16 - 16 = 32 \quad (\text{Not } 0) \][/tex]
Option A fails at checking the zeros.
### Option B: [tex]\( f(x) = x^2 - 2x - 8 \)[/tex]
1. Check the zeros:
[tex]\[ f(-2) = (-2)^2 - 2(-2) - 8 = 4 + 4 - 8 = 0 \][/tex]
[tex]\[ f(4) = 4^2 - 2(4) - 8 = 16 - 8 - 8 = 0 \][/tex]
Both zeros are correct.
2. Check the [tex]\( y \)[/tex]-intercept:
[tex]\[ f(0) = 0^2 - 2(0) - 8 = -8 \quad (\text{Not } -16) \][/tex]
Option B seems promising but does not match the [tex]\( y \)[/tex]-intercept.
### Option C: [tex]\( f(x) = x^2 + 2x - 8 \)[/tex]
1. Check the zeros:
[tex]\[ f(-2) = (-2)^2 + 2(-2) - 8 = 4 - 4 - 8 = -8 \quad (\text{Not } 0) \][/tex]
[tex]\[ f(4) = 4^2 + 2(4) - 8 = 16 + 8 - 8 = 16 \quad (\text{Not } 0) \][/tex]
Option C fails at checking the zeros.
### Option D: [tex]\( f(x) = 2x^2 - 4x - 16 \)[/tex]
1. Check the zeros:
[tex]\[ f(-2) = 2(-2)^2 - 4(-2) - 16 = 8 + 8 - 16 = 0 \][/tex]
[tex]\[ f(4) = 2(4)^2 - 4(4) - 16 = 32 - 16 - 16 = 0 \][/tex]
Both zeros are correct.
2. Check the [tex]\( y \)[/tex]-intercept:
[tex]\[ f(0) = 2(0)^2 - 4(0) - 16 = -16 \][/tex]
[tex]\( y \)[/tex]-intercept is correct.
### Final Decision:
Based on the given conditions and the evaluations, Option D: [tex]\( f(x) = 2x^2 - 4x - 16 \)[/tex] is the function that represents a parabola with zeros at [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex], and a [tex]\( y \)[/tex]-intercept at [tex]\( (0, -16) \)[/tex].
Condition 1: Zeros at [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex]
A parabola with zeros at [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex] can be written in the factored form as:
[tex]\[ f(x) = a(x + 2)(x - 4) \][/tex]
Condition 2: [tex]\( y \)[/tex]-intercept at [tex]\( (0, -16) \)[/tex]
This means that when [tex]\( x = 0 \)[/tex], [tex]\( f(x) = -16 \)[/tex]. Using this information, we can test each given function.
### Option A: [tex]\( f(x) = 2x^2 + 4x - 16 \)[/tex]
1. Check the zeros:
[tex]\[ f(-2) = 2(-2)^2 + 4(-2) - 16 = 8 - 8 - 16 = -16 \quad (\text{Not } 0) \][/tex]
[tex]\[ f(4) = 2(4)^2 + 4(4) - 16 = 32 + 16 - 16 = 32 \quad (\text{Not } 0) \][/tex]
Option A fails at checking the zeros.
### Option B: [tex]\( f(x) = x^2 - 2x - 8 \)[/tex]
1. Check the zeros:
[tex]\[ f(-2) = (-2)^2 - 2(-2) - 8 = 4 + 4 - 8 = 0 \][/tex]
[tex]\[ f(4) = 4^2 - 2(4) - 8 = 16 - 8 - 8 = 0 \][/tex]
Both zeros are correct.
2. Check the [tex]\( y \)[/tex]-intercept:
[tex]\[ f(0) = 0^2 - 2(0) - 8 = -8 \quad (\text{Not } -16) \][/tex]
Option B seems promising but does not match the [tex]\( y \)[/tex]-intercept.
### Option C: [tex]\( f(x) = x^2 + 2x - 8 \)[/tex]
1. Check the zeros:
[tex]\[ f(-2) = (-2)^2 + 2(-2) - 8 = 4 - 4 - 8 = -8 \quad (\text{Not } 0) \][/tex]
[tex]\[ f(4) = 4^2 + 2(4) - 8 = 16 + 8 - 8 = 16 \quad (\text{Not } 0) \][/tex]
Option C fails at checking the zeros.
### Option D: [tex]\( f(x) = 2x^2 - 4x - 16 \)[/tex]
1. Check the zeros:
[tex]\[ f(-2) = 2(-2)^2 - 4(-2) - 16 = 8 + 8 - 16 = 0 \][/tex]
[tex]\[ f(4) = 2(4)^2 - 4(4) - 16 = 32 - 16 - 16 = 0 \][/tex]
Both zeros are correct.
2. Check the [tex]\( y \)[/tex]-intercept:
[tex]\[ f(0) = 2(0)^2 - 4(0) - 16 = -16 \][/tex]
[tex]\( y \)[/tex]-intercept is correct.
### Final Decision:
Based on the given conditions and the evaluations, Option D: [tex]\( f(x) = 2x^2 - 4x - 16 \)[/tex] is the function that represents a parabola with zeros at [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex], and a [tex]\( y \)[/tex]-intercept at [tex]\( (0, -16) \)[/tex].
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.
