How do you graph #y=|2x+3|#?

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We are given the absolute value function:

#color(red)(y=f(x)=|2x+3|#

The #color(blue)("General Form of an Absolute Value Function"#:

#color(green)(y=f(x)=a|mx-h|+k#, where

#color(green)(a, m, h, k in RR#

Vertex: #color(blue)(((h)/m,k)#

Axis of Symmetry: #color(blue)(x=(h)/m#

We are given

#color(red)(y=f(x)=|2x+3|#

#a=1; m=2; h=-3 and k = 0#

Note that #h=(-3)#, since the formula contains #(-h)#

Vertex : #color(red)(((-3)/2, 0)#

Hence, Vertex is #color(red)((-1.5, 0)#

Axis of Symmetry : #color(red)(((-3)/2)#

Hence, Axis of Symmetry is at #color(red)(x=(-1.5)#

Create a data table for the Parent Function

#y = f(x)=|x|# and

the given absolute value function

#y = f(x)=|2x+3|#

The data table is given below:

Draw the graph for #color(red)(y=f(x)=|x|#

Draw the graph for #color(red)(y=f(x)=|2x+3|#

Observe that,

Vertex :#color(red)((-1.5,0)#

Draw the Axis of Symmetry on the graph as shown below:

Keep both the graphs of

#color(blue)(y=f(x)=|x|# and

#color(blue)(y=f(x)=|2x+3|#

as shown below and analyze the transformations:

Observe that the values #color(blue)(a, m, h and k# influence the corresponding transformations.

All transformations are with reference to the parent graph.

#color(red)((h)# is responsible for a horizontal transformation.

#color(red)[[h=(-3)]# indicates that the graph shifts horizontally by 3 units to the left.

Hope you find this solution helpful.