Question

Graph each function 

17. f(x)=2|x-1|-2 

18. f(x)= x+3 if x<0 
2x if x > or equal to 0 

Graph each inequality 

19. -2< or equal to x-2y < or equal to 4 

20. y<-|x+1|+2 

Answer 1

18)The piecewise function

The break between two halves of the function is at x = 0

x	y=x+3	(x, y)
-3	y=-3+3=0	(-3,0)
-2	y=-2+3=1	(-2,1)
-1	y=-1+3=2	(-1,2)
0	y=0+3=3	(0,3)

 

x	y=2x	(x, y)
0	y=2(0)=0	(0, 0)
-1	y=2(1)=2	(1, 2)
2	y=2(2)=4	(2, 4)
3	y=2(3)=6	(3,6)

Graph

1.Draw the coordinate plane.

2.Plot the points found in the table.

3.Connect the plotted points.

In the graph the open circle means the point does not including and closed circle means the point is including.

.

Answer 2

17) The function f(x) = 2 | x - 1| - 2

Make a table

x	y = 2|x-1| - 2	(x, y)
-3	y=2 |-3-1|-2=6	(-3,6)
-2	y=2 |-2-1|-2=4	(-2,4)
-1	y=2 |-1-1|-2=2	(-1,2)
0	y=2|0-1|-2=0	(0,0)
1	y=2|1-1|-2=-2	(1,-2)
2	y=2|2-1|-2=0	(2,0)
3	y=2|3-1|-2=2	(3,2)
4	y=2|4-1|-2=4	(4,4)

Graph

1.Draw the coordinate plane.

2.Plot the points found in the table.

3.Connect the plotted points.

Answer 3

The absolute inequality  y < - | x + 1| + 2

The graph of the inequality y < - | x + 1| + 2 is the shaded region, so every point in the shaded region satisfies the inequality.

The graph of the equation y = - | x + 1| + 2 is the boundary of the region. Since the inequality symbol is <, the boundary is drawn as a dotted curve to show that points on the line does not satisfy the inequality and the shaded region of the graph of y = - | x + 1| + 2 is the soutions to the inequaity.

Write the equality y = - | x + 1| + 2

The sign is negative so the graph looks like down ward V.

Find the vertex of the graph  y = - | x + 1| + 2.

- x - 1  = 0

x = -1

For  x = -1 , y = - | -1 + 1| + 2

y = 2

The vertex of the graph is ( - 1, 2).

To graph the equation, we need some more points to make more accurate graph.

x	y = -|x+1|+2	(x, y )
-6	y = -|-6+1|+2 = -3	(- 6,- 3)
-4	y = -|-4+1|+2 = 1	(- 4, -1)
-2	y = -|-2+1|+2 = 1	(-2, 1)
-1	y = -|-1+1|+2 = 2	(-1, 2)
1	y = -|1+1|+2 = 0	(1, 0)
2	y = -|2+1|+2 = -1	(2, - 1)
4	y = -|4+1|+2 = -3	(4, - 3)

 

Answer 4

Contd ...

To draw inequality y < - | x + 1| + 2 follow the steps.

1.  Draw a coordinate plane.

2.  Plot the points and connect these points.

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

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

y < - | x + 1| + 2

 0 < - |0 + 1| + 2

0 < 1

Above statement is true.

Since the above statement is true, shade the region that contains the point (0,0).

 

Answer 5

19) The inequality - 2 ≤ x - 2y ≤ 4

The inequalities are - 2 ≤ x - 2y and x - 2y ≤ 4.

1) Draw the coordinate plane.

Now first inequality - 2 ≤ x - 2y.

2) Graph the line y = x/2 + 1

3) Since the inequality symbol is ≤ the boundary is included the solution set.

Graph the boundary of the inequality - 2 ≤ x - 2y with solid line.

4) To determine which half plane to be shaded use a test point in either half- plane.

A simple chioce is (0,0). Substitute x  = 0 and y  = 0 in original inequality

- 2 ≤ x - 2y

- 2 ≤ 0 - 2(0)

- 2 ≤ 0

The statement is true.

5) Since the statement is true , shade the region contain point (0,0)

Shaded in fuchsia colour.

Now second inequality x - 2y ≤ 4.

6) Graph the line y = x/2 - 2

7) Since the inequality symbol is ≤ the boundary is not included the solution set.

Graph the boundary of the inequality x - 2y ≤ 4 with solid line.

8) To determine which half plane to be shaded use a test point in either half- plane.

A simple chioce is (0,0). Substitute x  = 0 and y  = 0 in original inequality

0 - 2(0) ≤ 4

0 ≤ 4

The statement is true.

9) Since the statement is true , shade the region contain point (0,0)

shaded in aqua colour.

The solution of the system is the set of ordered pairs in the intersection of the graph of - 2 ≤ x - 2y and x - 2y ≤ 4. This region is shaded in light purple color.