Answer:
[tex]a=-\dfrac{1}{2}[/tex]
Step-by-step explanation:
Given the parabola
[tex]y=a(x-1)(x-4)[/tex]
To find y-intercept, substitute [tex]x=0:[/tex]
[tex]y=a(0-1)(0-4)\\ \\y=4a[/tex]
Hence, [tex](0,-4a)[/tex] is the point of intersection with y-axis.
To find x-intercept, substitute [tex]y=0:[/tex]
[tex]0=a(x-1)(x-4)\\ \\x_1=1\ \text{or}\ x_2=4[/tex]
Hence, points [tex](1,0)[/tex] and [tex](4,0)[/tex] are the points of intersection with x-axis.
Find the area of the triangle formed:
[tex]A_{\triangle}=\dfrac{1}{2}\cdot \text{Base}\cdot \text{Height}\\ \\\text{Base}=|4-1|=3\\ \\\text{Height}=|4a|\\ \\A_{\triangle }=\dfrac{1}{2}\cdot 3\cdot |4a|=6|a|[/tex]
Since the area of the triangle is 3, you have
[tex]6|a|=3\\ \\6a=-3\ [\text{Parabola goes downward}]\\ \\a=-\dfrac{1}{2}[/tex]
