Complete the table of values for $y=3 x^2+4$.
What numbers replace A and B?
$x$ | -3 | -2 | -1 | 0 | 1 |
$y$ | 31 | A | 7 | B | 7 |
The problem requires finding the missing values \(A\) and \(B\) in a table of values for the quadratic function \(y = 3x^2 + 4\). The given table relates input values \(x\) to output values \(y\), with \(A\) corresponding to \(x = -2\) and \(B\) corresponding to \(x = 0\). To solve this, we must substitute these specific \(x\)-values into the given function and calculate the resulting \(y\)-values.
To find the value of \(A\), we substitute \(x = -2\) into the function \(y = 3x^2 + 4\):
\[A = 3(-2)^2 + 4\]
First, calculate the square of \(-2\):
\[(-2)^2 = 4\]
Then, multiply by 3 and add 4:
\[A = 3(4) + 4 = 12 + 4 = 16\]
Thus, the value of \(A\) is 16.
To find the value of \(B\), we substitute \(x = 0\) into the function \(y = 3x^2 + 4\):
\[B = 3(0)^2 + 4\]
First, calculate the square of 0:
\[0^2 = 0\]
Then, multiply by 3 and add 4:
\[B = 3(0) + 4 = 0 + 4 = 4\]
Thus, the value of \(B\) is 4.
We have determined that for the function \(y = 3x^2 + 4\), the missing values in the table are \(A = 16\) and \(B = 4\). Therefore, the table is completed as follows: \(A = 16\) and \(B = 4\). The final values are \(A = \boxed{16}\) and \(B = \boxed{4}\).
Quadratic functions are widely used in physics and engineering to model parabolic paths. For instance, if you are designing a fountain where water streams follow a parabolic arc, you can use a quadratic equation to predict the height of the water at various distances from the base. By substituting different horizontal distances (\(x\)) into your equation, you can determine exactly how high the water will be at those points, just as we calculated the \(y\)-values for our table. This allows for precise planning of structures, ensuring that the water lands exactly where intended.