Graph f(x)=x^2|x|

$f (x) = x^{2} | x |$

Find the absolute value vertex. In this case, the vertex for $y = x^{2} | x |$ is $(0, 0)$ .

To find the $x$ coordinate of the vertex, set the inside of the absolute value $x$ equal to $0$ . In this case, $x = 0$ .

$x = 0$

Replace the variable $x$ with $0$ in the expression.

$y = {(0)}^{2} | 0 |$

Simplify ${(0)}^{2} | 0 |$ .

Raising $0$ to any positive power yields $0$ .

$y = 0 | 0 |$

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$y = 0 \cdot 0$

Multiply $0$ by $0$ .

$y = 0$

The absolute value vertex is $(0, 0)$ .

$(0, 0)$

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

For each $x$ value, there is one $y$ value. Select a few $x$ values from the domain. It would be more useful to select the values so that they are around the $x$ value of the absolute value vertex.

Substitute the $x$ value $- 2$ into $f (x) = x^{2} | x |$ . In this case, the point is $(- 2, 8)$ .

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = {(- 2)}^{2} | - 2 |$

Simplify the result.

Raise $- 2$ to the power of $2$ .

$f (- 2) = 4 | - 2 |$

The absolute value is the distance between a number and zero. The distance between $- 2$ and $0$ is $2$ .

$f (- 2) = 4 \cdot 2$

Multiply $4$ by $2$ .

$f (- 2) = 8$

The final answer is $8$ .

$y = 8$

Substitute the $x$ value $- 1$ into $f (x) = x^{2} | x |$ . In this case, the point is $(- 1, 1)$ .

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = {(- 1)}^{2} | - 1 |$

Simplify the result.

Raise $- 1$ to the power of $2$ .

$f (- 1) = 1 | - 1 |$

Multiply $| - 1 |$ by $1$ .

$f (- 1) = | - 1 |$

The absolute value is the distance between a number and zero. The distance between $- 1$ and $0$ is $1$ .

$f (- 1) = 1$

The final answer is $1$ .

$y = 1$

Substitute the $x$ value $2$ into $f (x) = x^{2} | x |$ . In this case, the point is $(2, 8)$ .

Replace the variable $x$ with $2$ in the expression.

$f (2) = {(2)}^{2} | 2 |$

Simplify the result.

Raise $2$ to the power of $2$ .

$f (2) = 4 | 2 |$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$f (2) = 4 \cdot 2$

Multiply $4$ by $2$ .

$f (2) = 8$

The final answer is $8$ .

$y = 8$

The absolute value can be graphed using the points around the vertex $(0, 0), (- 2, 8), (- 1, 1), (1, 1), (2, 8)$

$\begin{array}{c} x & y \\ - 2 & 8 \\ - 1 & 1 \\ 0 & 0 \\ 1 & 1 \\ 2 & 8 \end{array}$

Graph f(x)=x^2|x|

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