Answer:
*Parabola that opens up
*y-intercept is 2
*x-intercepts are -1/3 and -2
*vertex is (-7/6 , -25/12)
Step-by-step explanation:
I will describe what this graph looks like.
First, the graph is quadratic because it is in the form ax² + bx + c. The function must be in the shape of a parabola. Parabolas look like a "U" shape.
Whether "a" is negative or positive represents of the parabola opens up or down. Since "3" is positive, the parabola opens up.
"c" represents the y-intercept. The y-intercept is when the graph touches the y-axis. The function must have a y-intercept of positive 2.
We can also find its x-intercepts, also called roots/zeroes, by substituting into the quadratic formula [tex]x=\frac{-b±\sqrt{b^{2}-4ac } }{2a}[/tex] (Ignore the Â).
Using the form ax² + bx + c, we know:
a=3; b=7; c=2
Substitute into the formula.
[tex]x=\frac{-b±\sqrt{b^{2}-4ac } }{2a}[/tex]
[tex]x=\frac{-7±\sqrt{7^{2}-4(3)(2) } }{2(3)}[/tex]
[tex]x=\frac{-7±\sqrt{25 } }{6}[/tex]
[tex]x=\frac{-7±5 }{6}[/tex]
Split the equation at the ± sign.
[tex]x=\frac{-7+5 }{6}[/tex]
[tex]x=\frac{-2}{6}[/tex]
[tex]x=-\frac{1}{3}[/tex]
[tex]x=\frac{-7-5 }{6}[/tex]
[tex]x=\frac{-12 }{6}[/tex]
[tex]x=-2[/tex]
The graph has x-intercepts -2 and -1/3.
We can find the vertex of the graph. It is the part of the parabola that is the lowest (or highest, is it opens down).
Find the midpoint of the x-intercepts for the vertex x-coordinate:
(-1/3 - 2)/2 = (-1/3 - 6/3)/2 = (-7/3)*(1/2) = -7/6 = -1.167 = x
Substitute the vertex x-coordinate into the formula to find the "y" in vertex.
y=3x²+7x+2
y=3(-7/6)²+7(-7/6)+2
y= 147/36 + (-49/6) + 2
y= 147/36 + (-294/36) + 72/36
y= (147-294+72)/36
y = -75/36
y = -25/12 = -2.083
The vertex is (-7/6 , -25/12) OR about (-1.167, - 2.083).
