Find the Vertex f(x)=3/2|x-2/5|+5/3

$f (x) = \frac{3}{2} | x - \frac{2}{5} | + \frac{5}{3}$

Step 1

To find the $x$ coordinate of the vertex, set the inside of the absolute value $x - \frac{2}{5}$ equal to $0$ . In this case, $x - \frac{2}{5} = 0$ .

$x - \frac{2}{5} = 0$

Step 2

Add $\frac{2}{5}$ to both sides of the equation.

$x = \frac{2}{5}$

Step 3

Replace the variable $x$ with $\frac{2}{5}$ in the expression.

$y = \frac{3}{2} \cdot | (\frac{2}{5}) - \frac{2}{5} | + \frac{5}{3}$

Step 4

Simplify $\frac{3}{2} \cdot | (\frac{2}{5}) - \frac{2}{5} | + \frac{5}{3}$ .

Simplify each term.

Combine the numerators over the common denominator.

$y = \frac{3}{2} \cdot | \frac{2 - 2}{5} | + \frac{5}{3}$

Subtract $2$ from $2$ .

$y = \frac{3}{2} \cdot | \frac{0}{5} | + \frac{5}{3}$

Remove non-negative terms from the absolute value.

$y = \frac{3}{2} \cdot \frac{0}{5} + \frac{5}{3}$

Divide $0$ by $5$ .

$y = \frac{3}{2} \cdot 0 + \frac{5}{3}$

Multiply $\frac{3}{2}$ by $0$ .

$y = 0 + \frac{5}{3}$

Add $0$ and $\frac{5}{3}$ .

$y = \frac{5}{3}$

Step 5

The absolute value vertex is $(\frac{2}{5}, \frac{5}{3})$ .

$(\frac{2}{5}, \frac{5}{3})$

Step 6