Graph y=x^2+4x+6

Problem

y=x2+4*x+6

Solution

1. Identify the type of function and its orientation. The equation is a quadratic function in the form y=a*x2+b*x+c where a=1 b=4 and c=6 Since a>0 the parabola opens upward.

2. Find the vertex using the formula x=−b/(2*a)

x=−4/(2*(1))=−2

3. Calculate the y-coordinate of the vertex by substituting x=−2 into the original equation.

y=(−2)2+4*(−2)+6

y=4−8+6=2

The vertex is at (−2,2)

4. Determine the y-intercept by setting x=0

y=(0)2+4*(0)+6=6

The y-intercept is at (0,6)

5. Find additional points to improve the accuracy of the graph. Choose x=−1 and x=−3

y=(−1)2+4*(−1)+6=1−4+6=3

y=(−3)2+4*(−3)+6=9−12+6=3

The points are (−1,3) and (−3,3)

6. Check for x-intercepts by setting y=0 and calculating the discriminant D=b2−4*a*c

D=4−4*(1)*(6)=16−24=−8

Since the discriminant is negative, there are no real x-intercepts.

Final Answer

y=(x+2)2+2

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