IDNLearn.com: Your go-to resource for finding precise and accurate answers. Get timely and accurate answers to your questions from our dedicated community of experts who are here to help you.
Sagot :
To determine which quadratic equation fits the data presented in the table, we will test each given equation against the provided data points.
Here are the given equations:
1. [tex]\( y = x^2 + x + 3 \)[/tex]
2. [tex]\( y = -x^2 - x - 3 \)[/tex]
3. [tex]\( y = x^2 - x + 3 \)[/tex]
4. [tex]\( y = x^2 - x - 3 \)[/tex]
And the data points are:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -5 & 33 \\ \hline -2 & 9 \\ \hline -1 & 5 \\ \hline 0 & 3 \\ \hline 3 & 9 \\ \hline 4 & 15 \\ \hline 6 & 33 \\ \hline \end{array} \][/tex]
We will evaluate each equation step by step:
### 1. Testing [tex]\( y = x^2 + x + 3 \)[/tex]:
For [tex]\( x = -5 \)[/tex]:
[tex]\[ y = (-5)^2 + (-5) + 3 = 25 - 5 + 3 = 23 \quad (\text{Given: } 33) \][/tex]
Since 23 ≠ 33, this equation does not fit.
### 2. Testing [tex]\( y = -x^2 - x - 3 \)[/tex]:
For [tex]\( x = -5 \)[/tex]:
[tex]\[ y = -(-5)^2 - (-5) - 3 = -25 + 5 - 3 = -23 \quad (\text{Given: } 33) \][/tex]
Since -23 ≠ 33, this equation does not fit.
### 3. Testing [tex]\( y = x^2 - x + 3 \)[/tex]:
For [tex]\( x = -5 \)[/tex]:
[tex]\[ y = (-5)^2 - (-5) + 3 = 25 + 5 + 3 = 33 \quad (\text{Given: } 33) \][/tex]
For [tex]\( x = -2 \)[/tex]:
[tex]\[ y = (-2)^2 - (-2) + 3 = 4 + 2 + 3 = 9 \quad (\text{Given: } 9) \][/tex]
For [tex]\( x = -1 \)[/tex]:
[tex]\[ y = (-1)^2 - (-1) + 3 = 1 + 1 + 3 = 5 \quad (\text{Given: } 5) \][/tex]
For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = (0)^2 - 0 + 3 = 0 - 0 + 3 = 3 \quad (\text{Given: } 3) \][/tex]
For [tex]\( x = 3 \)[/tex]:
[tex]\[ y = (3)^2 - 3 + 3 = 9 - 3 + 3 = 9 \quad (\text{Given: } 9) \][/tex]
For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = (4)^2 - 4 + 3 = 16 - 4 + 3 = 15 \quad (\text: } 15) \][/tex]
For [tex]\( x = 6 \)[/tex]:
[tex]\[ y = (6)^2 - 6 + 3 = 36 - 6 + 3 = 33 \quad (\text{Given: } 33) \][/tex]
Since all the calculations match the provided data points, this equation fits.
### 4. Testing [tex]\( y = x^2 - x - 3 \)[/tex]:
For [tex]\( x = -5 \)[/tex]:
[tex]\[ y = (-5)^2 - (-5) - 3 = 25 + 5 - 3 = 27 \quad (\text{Given: } 33) \][/tex]
Since 27 ≠ 33, this equation does not fit.
Thus, the quadratic equation that fits the given data is:
[tex]\[ y = x^2 - x + 3 \][/tex]
So the correct answer is:
[tex]\( y = x^2 - x + 3 \)[/tex] which corresponds to option 3.
Here are the given equations:
1. [tex]\( y = x^2 + x + 3 \)[/tex]
2. [tex]\( y = -x^2 - x - 3 \)[/tex]
3. [tex]\( y = x^2 - x + 3 \)[/tex]
4. [tex]\( y = x^2 - x - 3 \)[/tex]
And the data points are:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -5 & 33 \\ \hline -2 & 9 \\ \hline -1 & 5 \\ \hline 0 & 3 \\ \hline 3 & 9 \\ \hline 4 & 15 \\ \hline 6 & 33 \\ \hline \end{array} \][/tex]
We will evaluate each equation step by step:
### 1. Testing [tex]\( y = x^2 + x + 3 \)[/tex]:
For [tex]\( x = -5 \)[/tex]:
[tex]\[ y = (-5)^2 + (-5) + 3 = 25 - 5 + 3 = 23 \quad (\text{Given: } 33) \][/tex]
Since 23 ≠ 33, this equation does not fit.
### 2. Testing [tex]\( y = -x^2 - x - 3 \)[/tex]:
For [tex]\( x = -5 \)[/tex]:
[tex]\[ y = -(-5)^2 - (-5) - 3 = -25 + 5 - 3 = -23 \quad (\text{Given: } 33) \][/tex]
Since -23 ≠ 33, this equation does not fit.
### 3. Testing [tex]\( y = x^2 - x + 3 \)[/tex]:
For [tex]\( x = -5 \)[/tex]:
[tex]\[ y = (-5)^2 - (-5) + 3 = 25 + 5 + 3 = 33 \quad (\text{Given: } 33) \][/tex]
For [tex]\( x = -2 \)[/tex]:
[tex]\[ y = (-2)^2 - (-2) + 3 = 4 + 2 + 3 = 9 \quad (\text{Given: } 9) \][/tex]
For [tex]\( x = -1 \)[/tex]:
[tex]\[ y = (-1)^2 - (-1) + 3 = 1 + 1 + 3 = 5 \quad (\text{Given: } 5) \][/tex]
For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = (0)^2 - 0 + 3 = 0 - 0 + 3 = 3 \quad (\text{Given: } 3) \][/tex]
For [tex]\( x = 3 \)[/tex]:
[tex]\[ y = (3)^2 - 3 + 3 = 9 - 3 + 3 = 9 \quad (\text{Given: } 9) \][/tex]
For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = (4)^2 - 4 + 3 = 16 - 4 + 3 = 15 \quad (\text: } 15) \][/tex]
For [tex]\( x = 6 \)[/tex]:
[tex]\[ y = (6)^2 - 6 + 3 = 36 - 6 + 3 = 33 \quad (\text{Given: } 33) \][/tex]
Since all the calculations match the provided data points, this equation fits.
### 4. Testing [tex]\( y = x^2 - x - 3 \)[/tex]:
For [tex]\( x = -5 \)[/tex]:
[tex]\[ y = (-5)^2 - (-5) - 3 = 25 + 5 - 3 = 27 \quad (\text{Given: } 33) \][/tex]
Since 27 ≠ 33, this equation does not fit.
Thus, the quadratic equation that fits the given data is:
[tex]\[ y = x^2 - x + 3 \][/tex]
So the correct answer is:
[tex]\( y = x^2 - x + 3 \)[/tex] which corresponds to option 3.
We are happy to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. For precise answers, trust IDNLearn.com. Thank you for visiting, and we look forward to helping you again soon.
