helppp! 8

Question

Sketch 3 different graphs that show you know the different number of x intercepts that are possible for a quadratic function.

 

Answer 1

The quadratic function y = ax² + bx + c

A quadratic function represents a parabola in the graph.

Depending up on the Discriminant(b² - 4ac) values we determine the possible x intercepts.

Case 1 :

Example equation y = 2x² + x + 2

Compare it to y = ax² + bx + c.

a = 2, b = 1, c = 2

b² - 4ac = (1)² - 4(2)(2)

= 1 - 16 = - 15

Discriminant b² - 4ac is negative means the equation has two imaginary roots.

Therefore, the graph of parabola never touches the x - axis.

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

x	y = 2x2 + x + 2	(x, y)
-1	y = 2(-1)2 - 1 + 2	(-1, 3)
-2	y = 2(-2)2 - 2 + 2	(- 2, 8)
0	y = 2(0)2 + 0 + 2	(0, 2)
1	y = 2(1)2 + 1 + 2	(1, 5)
2	y = 2(2)2 + 2 + 2	(2, 12)

1.Draw a coordinate plane.

2.Plot the coordinate points.

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

Observe the graph there is no x intercepts.

Answer 2

Case 2 :

Example equation y = - 2x² + x + 1

Compare it to y = ax² + bx + c.

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

b² - 4ac = (1)² - 4(- 2)(1)

= 1 + 8 = 9

Discriminant b² - 4ac is positive means the equation has two real roots.

Therefore, the graph of parabola touches the x - axis at two points.

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

x	y = - 2x2 + x + 1	(x, y)
-2	y = - 2(-2)2 - 2 + 1	(- 1, - 2)
-1	y = - 2(-1)2 - 1 + 1	(- 2, - 9)
0	y = - 2(0)2 + 0 + 1	(0, 1)
1	y = - 2(1)2 + 1 + 1	(1, 0)
2	y = - 2(2)2 + 2 + 1	(2, - 5)

1.Draw a coordinate plane.

2.Plot the coordinate points.

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

Observe the graph, there is two x intercepts of parabola.

Answer 3

Case 3 :

Example equation y = x² + 4x + 4

Compare it to y = ax² + bx + c.

a = 1, b = 4, c = 4

b² - 4ac = (4)² - 4(1)(4)

= 16 - 16 = 0

Discriminant b² - 4ac is 0 means the equation has only one real root.

Therefore, the graph of parabola touches the x - axis at one point.

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

x	y = x2 + 4x + 4	(x, y)
-4	y = (-4)2+ 4(-4)+ 4	(- 4, 4)
-2	y = (-2)2+ 4(-2)+ 4	(- 2, 0)
-1	y = (-1)2+ 4(-1)+ 4	(-1, 1)
0	y = (0)2+ 4(0)+ 4	(0, 4)
0.5	y =(0.5)2+4(0.5)+ 4	(0.5, 6.25)

1.Draw a coordinate plane.

2.Plot the coordinate points.

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

Observe the graph, there is only one x intercept of parabola.