Another Quadratic Inequality

Question

Sketch the graph of y ≤ - x² - 5x + 4

Answer 1

Given quadratic inequality y < = -x^2 - 5x + 4

Let x = -1=> y = -(-1)^2 - 5 ( -1) + 4 => y = 8

Let x = -2=> y = -(-2)^2 - 5 ( -2) + 4 => y = 10

Let x = -4=> y = -(-4)^2 - 5 ( -4) + 4 => y = 8

1) Draw a smooth curve for y = -x^2 - 5x + 4.

2) Plot the points on the curve.

3) Therefore the required in equality graph is

Comments

This is inequality problem. So, area under curve need to be shaded.

Answer 2

The quadratic inequality y  ≤ -x ² - 5x + 4

Draw the coordinate plane.

The fact that these expression is inequality and not equation doesn’t change the general shape of the graph at all.

y = -x ² - 5x + 4

Compare it to parabola equaion y  =  a x ² + b x + c.

a  = -1, b  = -5 , c  = 4

a  is negative so the parabola opens downward.

Since the inequality symbol is ≤ so the boundary is included the solution set.

Graph the boundary of the parabola with solid line.

To determine which  plane to be shaded use a test point in either plane.

A Simple choice is (0,0) substitute x  = 0 and y  = 0 in original inequality.

0 ≤ - 0² - 5(0) + 4

0 ≤ 4

The above statement is true.

So shade inside the parabola.

Answer 3

The inequality is y ≤ - x² - 5x +4.

Write the equality is y = - x² - 5x + 4 and it is represent a parabola curve.

The graph of the inequality y ≤ - x² - 5x +4 is the shaded region, so every point in the shaded region satisfies the inequality.

The graph of the equation y = - x² - 5x + 4 is the boundary of the region. Since the inequality symbol is ≤, the boundary is drawn as a solid curve to show that points on the curve does satisfy the inequality.

To graph the boundary curve make the table.

Make the table of values to find ordered pairs that satisfy the equation.

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

x	y = - x2 - 5x + 4	(x, y)
- 7	y = - (- 7)2 - 5(- 7) + 4 = 39 - 49 = - 10	(- 7, - 10)
- 6	y = - (- 6)2 - 5(- 6) + 4 = 34 - 36 = - 2	(- 6, - 2)
- 5	y = - (- 5)2 - 5(- 5) + 4 = 4	(- 5, 4)
- 4	y = - (- 4)2 - 5(- 4) + 4 = 24 - 16 = 8	(- 4, 8)
- 2	y = - (- 2)2 - 5(- 2) + 4 = 10	(- 2, 10)
0	y = - (0)2 - 5(0) + 4 = 4	(0, 4)
1	y = - (1)2 - 5(1) + 4 = - 1 - 5 + 4 = - 2	(1, - 2)
2	y = - (2)2 - 5(2) + 4 = - 10	(2, - 10)

To draw inequality y ≤ - x² - 5x +4 follow the steps.

1.  Draw a coordinate plane.

2.  Plot the points and draw a smooth curve through these points.

3.  To determine which side (out side or in side) to be shaded, use a test point inside the parabola. A simple choice is (0, 0).

Substitute the value of (x, y) = (0, 0) in the original inequality.

0 ≤ - 0² - 5(0) + 4

0 ≤ 4.

4.  Since the above statement is true, shade the region inside the parabola.