The correct graph is fourth option
Explanation:[tex]\begin{gathered} \text{The quadratic function:} \\ \text{y = }\frac{-1}{2}x^2\text{ - }\frac{3}{2}x\text{ + 2} \end{gathered}[/tex]Using the intercepts: we have x-intercept and y-intercept
To get x intercept, we will equate the function to zero. That is y = 0
[tex]\begin{gathered} 0\text{ = }\frac{-1}{2}x^2\text{ - }\frac{3}{2}x\text{ + 2} \\ mu\text{ltiply through by 2:} \\ 0=-x^2\text{ - 3x + 4} \\ x^2\text{ + 3x - 4 = 0} \\ x^2\text{ - x + 4x - 4 = 0} \\ x(x\text{- 1) +4(x - 1) = 0} \\ (x\text{ + 4)(x - 1) = 0} \\ x+4\text{ = 0 or x - 1 = 0} \\ x\text{ = -4 or x = 1} \end{gathered}[/tex]x intercepts are x = -4 and x = 1
This means the line will cross the x axis at two points: x = -4 and x = 1
To get y-intercept, we will equate x to zero:
[tex]\begin{gathered} \text{y = }\frac{-1}{2}(0)^2\text{ - }\frac{3}{2}(0)\text{ + 2} \\ y\text{ = 0 - 0 + 2} \\ y\text{ = 2} \end{gathered}[/tex]y-intercept = 2
This means the line will cross the y axis at y = 2
The correct graph is fourth option