1. (7 points) On the coordinate plane, the graph of the function $y=a x^{2}+b x+c$ is shown (see figure). On the same coordinate plane, schematically draw the graph of the function $y=c x^{2}+2 b x+a$. Explain your answer.

![](https://cdn.mathpix.com/cropped/2024_05_06_23a1ae9974d265e1d2b8g-1.jpg?height=608&width=513&top_left_y=747&top_left_x=840)

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# Solution.

Let's calculate the values of $a, b$, and $c$.

Method 1. Choose three convenient points from the given graph, for example, $(0 ; 1)$; $(1 ;-2)$, and $(-1 ; 2)$. Considering that $y(0)=c, y(1)=a+b+c, y(-1)=a-b+c$, we get the system of equations

$$
\left\{\begin{array}{l}
c=1 \\
a+b+c=-2 \\
a-b+c=2
\end{array}\right.
$$

The solution to this system is: $a=-1, b=-2, c=1$.

Method 2. Notice that the given graph is obtained by a parallel shift of the graph of the function $y=-x^{2}$, so $a=-1$. The value of $b=-2$ is calculated from the equation $-\frac{b}{2 a}=-1$ (the x-coordinate of the vertex of the parabola), and $c=y(0)=1$.

Therefore, the desired graph is given by the equation $y=x^{2}-4 x-1 \Leftrightarrow y=(x-2)^{2}-5$.

![](https://cdn.mathpix.com/cropped/2024_05_06_23a1ae9974d265e1d2b8g-2.jpg?height=688&width=494&top_left_y=193&top_left_x=861)