Question

Need help asap!!!!!

Need to graph and know the points

Answer 1

See the attachment for the graph of the piecewise function.

Plot points (2, -2), (-4, -8) and (4, -6).

Draw a straight line from (2, -2) through (-4, -8).

Draw a straight line from (2, -2) through (4, -6).

Add arrows at the ends of the lines to show that they continue indefinitely in that direction.

Explanation:

Piecewise functions have multiple pieces of curves and/or lines, where each piece corresponds to its definition over an interval.

Given piecewise function:

`f(x)=\begin{cases}x-4\; &\text{if}\;\;x < 2\\-2x+2\; & \text{if}\;\;x\geq 2\end{cases}`

Therefore, the function has two definitions:

• f(x) = x - 4 when x is less than 2.
• f(x) = -2x + 2 when x is more than or equal to 2.

When graphing piecewise functions:

• Use an open circle where the endpoint is not included in the interval.
• Use a closed circle where the endpoint is included in the interval.
• Use an arrow to show that the function continues indefinitely in that direction.

First piece of the function

Substitute the endpoint x = 2 into the first function:

`f(2)=2-4=-2`

As x = 2 is not included in the interval, place an open circle at point (2, -2).

To help graph the line, find another point on the line by inputting another value of x that is less than 2 into the same function:

`f(-4)=-4-4=-8`

Plot point (-4, -8) and draw a straight line from the open circle at (2, -2) through point (-4, -8). Add an arrow at the other endpoint to show it continues indefinitely as x → -∞.

Second piece of the function

Substitute the endpoint x = 2 into the second function:

`f(2)=-2(2)+2=-2`

Notice that the endpoint of this piece of the function when x = 2 is the same as the endpoint of the first piece of the function when x = 2.

Therefore, we need to remove the open circle we placed earlier as (2, -2), as the function is continuous at x = 2.

To help graph the line of the second piece of the function, find another point on the line by inputting another value of x that is more than 2 into the same function:

`f(4)=-2(4)+2=-6`

Plot point (4, -6) and draw a straight line from (2, -2) through point (4, -6). Add an arrow at the other endpoint to show it continues indefinitely as x → ∞.

Answer 2

Graphing by plotting system.

In order to graph the function
`\tt f(x) = \begin{cases} x-4, & \text{if } x < 2 \\ -2x+2 & \text{if } x \geq 2 \end{cases}`

we'll follow the same steps as before.

1. For x < 2

The function is f(x) = x - 4 for values less than 2.

Let's choose three values for x to create the line segment:

x = 0: f(0) = 0 - 4 = -4 Point A(0, -4)

x = 1: f(1) = 1 - 4 = -3 Point B(1, -3)

x=2: f(2) is not defined in this case since the condition is x< 2.

2. For
`\tt x \geq 2`

The function becomes f(x) = -2x + 2 for x greater than or equal to 2.

Let's choose three values for x to create the line segment:

x = 2: f(2) = -2(2) + 2 = -2 Point C(2, -2)

x = 3: f(3) = -2(3) + 2 = -4 Point D(3, -4)

x = 4: f(4) = -2(4) + 2 = -6 Point E(4, -6)

Plot the above points and make a straight line.

The graph shows that two lines are joined at point C(2, -2)