2. In the function $f(x)=\frac{x^{2}+a}{x^{2}+b}$, the constant $a$ is equal to the abscissa, and $b$ is the ordinate of the point $M$, where the function $g(x)=2 x^{2}+12 x+19$ achieves its minimum value. Determine the constants $a$ and $b$ and sketch the graph of the function $f(x)$.
2. We note that $a$ is the abscissa, and $b$ is the ordinate of the vertex of the quadratic function. We calculate $a=$ $-\frac{b}{2 a}=-3$ and $b=-\frac{D}{4 a}=1$. We write the function $f(x)=\frac{x^{2}-3}{x^{2}+1}$. We determine the zeros $x_{1}=\sqrt{3}$ and $x_{2}=-\sqrt{3}$, there are no poles, the asymptote is $y=1$, and the initial value is $(0,-3)$. We consider the signs and draw the graph.

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-11.jpg?height=63&width=1639&top_left_y=1896&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-11.jpg?height=57&width=1639&top_left_y=1950&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-11.jpg?height=54&width=1642&top_left_y=2003&top_left_x=204)

Conclusion that there are no poles .................................................................................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-11.jpg?height=58&width=1639&top_left_y=2104&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-11.jpg?height=55&width=1642&top_left_y=2154&top_left_x=207)

Graph drawn ...........................................................................................................