Question/Answer Booklet
Page 20
16. TWO FUNCTIONS
Functions $f$ and $g$ are second-degree polynomial functions.
They are represented by two parabolas on the Cartesian plane below.
The following table of values represents function $f$.
\begin{tabular}{|c|c|}
\hline$x$ & $f(x)$ \\
\hline-1 & 57 \\
\hline 8 & 39 \\
\hline 9 & 57 \\
\hline
\end{tabular}
- $\operatorname{Ran} f=[7, \infty[$
$f(x)=g(x)=25$
The equation of the axis of symmetry of function $g$ is $x=0.5$.
One of the zeros of function $g$ is -4 .
What is the rule of function $g$ ?

Step 1 ：The rule of a second-degree polynomial function (or a quadratic function) is given by \(y = ax^2 + bx + c\). We know that the axis of symmetry of a quadratic function is given by \(x = -b/2a\). We are given that the axis of symmetry of function \(g\) is \(x = 0.5\), so we can set up the equation \(-b/2a = 0.5\) to solve for \(b\) in terms of \(a\).

Step 2 ：We are also given that one of the zeros of function \(g\) is -4. This means that when \(x = -4\), \(y = 0\). We can substitute these values into the equation \(y = ax^2 + bx + c\) to get another equation in terms of \(a\), \(b\), and \(c\).

Step 3 ：We can solve these two equations simultaneously to find the values of \(a\), \(b\), and \(c\), which will give us the rule of function \(g\).

Step 4 ：We have found that \(a = -0.05c\) and \(b = 0.05c\). However, we still need to find the value of \(c\). We know that \(f(x) = g(x) = 25\) for some \(x\), but we don't know the exact value of \(x\). Since \(f\) and \(g\) are both second-degree polynomial functions, they should have the same shape, and their graphs should be symmetric about their respective axes of symmetry. Therefore, the \(x\) value at which \(f(x) = g(x) = 25\) should be the same distance from the axis of symmetry of \(f\) as it is from the axis of symmetry of \(g\).

Step 5 ：We know that the axis of symmetry of \(f\) is the line \(x = (8 + (-1))/2 = 3.5\), and the axis of symmetry of \(g\) is the line \(x = 0.5\). Therefore, the \(x\) value at which \(f(x) = g(x) = 25\) should be \(3.5 - (3.5 - 0.5) = -2.5\).

Step 6 ：We can substitute \(x = -2.5\) and \(y = 25\) into the equation \(y = ax^2 + bx + c\) to get a third equation in terms of \(a\), \(b\), and \(c\). We can then solve this equation along with the two equations we found earlier to find the values of \(a\), \(b\), and \(c\).

Step 7 ：Final Answer: The rule of function \(g\) is \(\boxed{y = -2.222x^2 + 2.222x + 44.444}\).