9.2. On the coordinate plane, the graphs of a linear and a quadratic function are plotted (see figure). The equation of the linear function has the form $y=c x+2 c$ for some number $c$. Using the same parameter $c$, write the equation of the quadratic function and explain your solution.
Answer: $y=0.5 c(x+2)^{2}$.

Solution. Let's find the coordinates of the points of intersection of the graph of the linear function with the coordinate axes: ( $0 ; 2 \mathrm{c}$ ) and (-2; 0). The graph of the quadratic function (parabola) touches the $O X$ axis at the point ( $-2 ; 0$ ), hence its formula has the form $y=a(x+2)^{2}$. Since the parabola passes through the point ( $0 ; 2 \mathrm{c}$ ), substituting $x=0, y=2 c$ into the obtained formula, we get: $2 c=a \cdot 2^{2}$, from which $a=$ $0.5 c$. Thus, the required formula is: $y=0.5 c(x+2)^{2}$.

Evaluation criteria. "+" A complete and justified solution and the correct answer are provided

" $\pm$ " It is correctly and justified that $y=a(x+2)^{2}$, but

![](https://cdn.mathpix.com/cropped/2024_05_06_6320f12e2ab8bf3a3a08g-1.jpg?height=719&width=760&top_left_y=931&top_left_x=1136)
there is no further progress or a computational error is made in calculating the value of $a$

“Ғ" Only the correct answer is provided

"-" An incorrect solution is provided or it is absent