How do I graph this?

Question

Sketch the graph of the following piecewise function:
f(x) ={-2x+3, -4≤x<-1

           −2x + 3, −4 ≤ x < −1
           |2x − 1|, −1 ≤ x ≤ 2
            5, x = 3
            3x² − 1, x > 3}

Answer 1

Lets start with the interval    -4 ≤ x ≤ -1.  The function associated with this domain is linear.  So lets evaluate the endpoints of this piece.

 

f(-4) = -2(-4) + 3 = 11

f(-1) = -2(-1) + 3 = 5

 

In this interval, draw a straight line from the point (-4, 11) to (-1, 5).  Draw a closed circle at (-4, 11) and an open circle at (-1, 5).

 

 

For interval  -1 ≤ x ≤ 2:

 

f(-1) = | 2(-1) - 1 | = 3

f(0) = 1 f(1) = 1

f(2) = | 2(2) - 1 | = 3

 

Draw a straight line from the point (-1, 3) to (2, 3) in the order of the points given. You should get a V-shape curve in this interval.  Closed circle at (-1, 3) and closed circle at (2, 3).

 

Plot the point (3, 5).  This is a closed circle.

 

 

For interval   x>3:

 

This is a parabola, so we need to find several point here, if they fall within the domain.

 

f(3) = 3(3)² - 1 = 26

 

The x-intercepts, vertex, and y-intercept do not fall within this domain. So we just evaluate several more point on this domain.

 

f(4) = 3(4)² - 1 = 47

f(5) = 3(5)² - 1 = 74

f(6) = 3(6)² - 1 = 107

 

Plot the points (3, 26), (4, 47), (5, 74), (6, 107).  Open circle at (3, 26). Then after that, the right half of the parabola stretches forever where the remaining points are close circles.

 

 

Now take the information I provided you that translate it into a graph.