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

Jul 27, 2018

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#### Explanation:

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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 $h = \left(- 3\right)$, since the formula contains $\left(- h\right)$

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

$y = f \left(x\right) = | x |$ and

the given absolute value function

$y = f \left(x\right) = | 2 x + 3 |$

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.