Example 1. Plot the graph of the function $y=\frac{1}{2} x^{2}+2 x-1$.
Solution. Rewrite the given function as $y=\frac{1}{2}(x+2)^{2}-3$.
![](https://cdn.mathpix.com/cropped/2024_05_22_8f5be56c3f01f571fd60g-056.jpg?height=990&width=406&top_left_y=522&top_left_x=90)

Fig. 5.12

1) $\Gamma_{1}: y=x^{2}$ - parabola (Fig. 5.12, a) with vertex at point $O(0,0)$.
2) $\Gamma_{2}: \quad y=(x+2)^{2}-\quad$ parabola (Fig. $5.12, b$ ) with vertex at point $A(-2,0)$.
3) $\Gamma_{3}: y=\frac{1}{2}(x+2)^{2}$. Obtained from $\Gamma_{2}$ by compressing it by a factor of 2 towards $O x$ (Fig. 5.12, c).
4) $\Gamma_{4}: y=\frac{1}{2}(x+2)^{2}-3 . \Gamma_{3}$ is lowered by 3 units (Fig. $5.12, d$ ).