# How do you graph f(x) =abs(2x+3)?

Apr 29, 2018

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#### Explanation:

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color(green)("Step 1"

Construct a table of values with Input: color(red)(x

Values are obtained for the graphs:

$y = | x |$

$y = | 2 x |$

$y = | 2 x + 3 |$

You can now analyze how the computed values reflect visually on their respective graphs.

color(green)("Step 2"

Consider color(blue)(y=a|x-h|+k

This will help understand how transformations will work.

$V e r t e x : \left(h , k\right)$

Graph of $y = | x |$

This is the Parent Graph.

$V e r t e x : \left(0 , 0\right)$

color(green)("Step 3"

Graph of $y = | 2 x |$

$V e r t e x : \left(0 , 0\right)$

color(green)("Step 4"

Graph of $y = | 2 x + 3 |$

Let us find the $V e r t e x :$

Let $b x - h = 0$

$\Rightarrow 2 x + 3 = 0$

Subtract 3 from both sides.

$\Rightarrow 2 x + \cancel{3} - \cancel{3} = 0 - 3$

$\Rightarrow 2 x = - 3$

Divide both sides by $2$

$\Rightarrow \frac{2 x}{2} = - \frac{3}{2}$

$\Rightarrow \frac{\cancel{2} x}{\cancel{2}} = - \frac{3}{2}$

$\Rightarrow x = - \frac{3}{2}$

$V e r t e x : \left(h . k\right)$

Hence, $V e r t e x : \left(- \frac{3}{2} , 3\right)$

Axis of Symmetry :$x = - \frac{3}{2} = - 1.5$

Domain is all possible x values: $\left(- \infty , \infty\right)$

Range: $\left[0 , \infty\right)$

Horizontal Shift:

$k = 3$ shifts $3$ units to the left.

Hope it helps.