Given the next quadratic function:
[tex]y=2x^2+3x-2[/tex]to sketch its graph, first, we need to find its vertex. The x-coordinate of the vertex is found as follows:
[tex]x_V=\frac{-b}{2a}[/tex]where a and b are the first two coefficients of the quadratic function. Substituting with a = 2 and b = 3, we get:
[tex]\begin{gathered} x_V=\frac{-3}{2\cdot2} \\ x_V=-\frac{3}{4}=-0.75 \end{gathered}[/tex]The y-coordinate of the vertex is found by substituting the x-coordinate in the quadratic function, as follows:
[tex]\begin{gathered} y_V=2x^2_V+3x_V-2 \\ y_V=2\cdot(-\frac{3}{4})^2+3\cdot(-\frac{3}{4})-2 \\ y_V=2\cdot\frac{9}{16}+3\cdot(-\frac{3}{4})-2 \\ y_V=\frac{9}{8}-\frac{9}{4}-2 \\ y_V=-\frac{25}{8}=-3.125 \end{gathered}[/tex]The factorization indicates that the curve crosses the x-axis at the points (-2, 0) and (1/2, 0). We also know that the curve crosses the y-axis at (0,-2). Connecting these points and the vertex (-0.75, -3.125) with a U-shaped curve, we get:
