Draw the Graph of y=2x^2-x-6

Question

a) find the y intercept b) find the X intercept c)Find the Vertex d)write the equation in the form y=a(x-h)^2+k?

Answer 1

The equation y = 2x² - x - 6

a) To find y intercept substitute x = 0 in y = 2x² - x - 6.

y = 2(0)² - 0 - 6

 y intercept is - 6.

b)To find x intercept substitute y = 0 in y = 2x² - x - 6

2x² - x - 6 = 0

2x² - 4x + 3x - 6 = 0

2x(x - 2) + 3(x - 2) = 0

(x - 2)(2x + 3) = 0

x - 2 = 0 and 2x = - 3

x = 2 and x = - 3/2

x intercepts are 2 and -3/2.

c) y = 2x² - x - 6

Compare it to y = ax² + bx + c

a = 2, b = - 1, c = - 6

To find vertex axis of symmetry x = - b/2a

x =  -(- 1)/ 2(2)

x = 1/4

To find y coordinate of vertex substitute x = 1/4 in y = 2x² - x - 6 .

y = 2(1/4)² - (1/4) - 6

y = 1/8 - 1/4 - 6

y = ( 1 - 2 - 48)/8

y = - 49/8

Vertex is (x, y) = (1/4 , -49/8) or ( 0.25, - 6.125).

Graph

Choose random values for y and find the corresponding values for x.

x	y = 2x2 - x - 6	(x, y)
1	y = 2(1)2 - 1 - 6	(1,- 5)
- 1	y = 2(-1)2 + 1 - 6	(-1, - 3)
- 2	y = 2(-2)2 + 2 - 6	(-2, 4)
2.5	y = 2(2.5)2 - 2.5 - 6	(7, - 3)

1.Draw a coordinate plane.

2.Plot the axis of symmetry x, y intercepts and coordinate points found in the table.

3.Then sketch the graph, connecting the points with a smooth curve.

Comments

Contd..

d) The equation is y = 2x² - x - 6.

Take out common factor.

y = 2(x² - (x/2) - 3)

To change the expression (x² - (x/2)) into a perfect square trinomial add (half the x coefficient)² to each side of the expression.

 Here x coefficient = - 1/2. So, (half the x coefficient)² = (- 1/4)² = 1/16.

Add and subtract 1/16 to the expression.

y = 2[x² - (x/2) - 3 + 1/16 - 1/16]

y = 2[x² - (x/2)  + 1/16 - 1/16 - 3]

y = 2[(x - 1/4)² - 49/16]

y = 2(x - 1/4)² - 49/8

Compare it to vertex form y = a(x - h)² + k

a = 2, h = 1/4, k = - 49/8.