Graph y=1/2x^2-x-6

Problem

y=1/2*x2−x−6

Solution

1. Identify the function type and its orientation. This is a quadratic function in the form y=a*x2+b*x+c where a=1/2 b=−1 and c=−6 Since a>0 the parabola opens upward.

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

x=−(−1)/(2*(1/2))=1

Substitute x=1 into the equation to find the ycoordinate:

y=1/2*(1)2−(1)−6=−6.5

The vertex is (1,−6.5)

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

y=1/2*(0)2−(0)−6=−6

The yintercept is (0,−6)

4. Find the x-intercepts by setting y=0 and solving the quadratic equation.

0=1/2*x2−x−6

Multiply the entire equation by 2 to clear the fraction:

0=x2−2*x−12

Apply the quadratic formula x=(−b±√(,b2−4*a*c))/(2*a)

x=(2±√(,(−2)2−4*(1)*(−12)))/(2*(1))

x=(2±√(,4+48))/2

x=(2±√(,52))/2=(2±2√(,13))/2=1±√(,13)

The xintercepts are approximately (4.61,0) and (−2.61,0)

5. Plot the points and draw a smooth U-shaped curve passing through the vertex (1,−6.5) the yintercept (0,−6) and the xintercepts.

Final Answer

y=1/2*x2−x−6

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