Calculus Examples

$y = \sqrt{x^{2} + x} - x$

Step 1

Find the $y$ value at $x = - 1$ .

Tap for more steps...

Step 1.1

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

$f (- 1) = \sqrt{(- 1) ((- 1) + 1)} - (- 1)$

Step 1.2

Simplify the result.

Tap for more steps...

Step 1.2.1

Simplify each term.

Tap for more steps...

Step 1.2.1.1

Add $- 1$ and $1$ .

$f (- 1) = \sqrt{- 1 \cdot 0} - (- 1)$

Step 1.2.1.2

Multiply $- 1$ by $0$ .

$f (- 1) = \sqrt{0} - (- 1)$

Step 1.2.1.3

Rewrite $0$ as $0^{2}$ .

$f (- 1) = \sqrt{0^{2}} - (- 1)$

Step 1.2.1.4

Pull terms out from under the radical, assuming positive real numbers.

$f (- 1) = 0 - (- 1)$

Step 1.2.1.5

Multiply $- 1$ by $- 1$ .

$f (- 1) = 0 + 1$

Step 1.2.2

Add $0$ and $1$ .

$f (- 1) = 1$

Step 1.2.3

The final answer is $1$ .

$1$

Step 1.3

The $y$ value at $x = - 1$ is $1$ .

$y = 1$

Step 2

Find the $y$ value at $x = - 2$ .

Tap for more steps...

Step 2.1

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

$f (- 2) = \sqrt{(- 2) ((- 2) + 1)} - (- 2)$

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Add $- 2$ and $1$ .

$f (- 2) = \sqrt{- 2 \cdot - 1} - (- 2)$

Step 2.2.1.2

Multiply $- 2$ by $- 1$ .

$f (- 2) = \sqrt{2} - (- 2)$

Step 2.2.1.3

Multiply $- 1$ by $- 2$ .

$f (- 2) = \sqrt{2} + 2$

Step 2.2.2

The final answer is $\sqrt{2} + 2$ .

$\sqrt{2} + 2$

Step 2.3

The $y$ value at $x = - 2$ is $\sqrt{2} + 2$ .

$y = \sqrt{2} + 2$

Step 3

Find the $y$ value at $x = - 3$ .

Tap for more steps...

Step 3.1

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = \sqrt{(- 3) ((- 3) + 1)} - (- 3)$

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Add $- 3$ and $1$ .

$f (- 3) = \sqrt{- 3 \cdot - 2} - (- 3)$

Step 3.2.1.2

Multiply $- 3$ by $- 2$ .

$f (- 3) = \sqrt{6} - (- 3)$

Step 3.2.1.3

Multiply $- 1$ by $- 3$ .

$f (- 3) = \sqrt{6} + 3$

Step 3.2.2

The final answer is $\sqrt{6} + 3$ .

$\sqrt{6} + 3$

Step 3.3

The $y$ value at $x = - 3$ is $\sqrt{6} + 3$ .

$y = \sqrt{6} + 3$

Step 4

Find the $y$ value at $x = 0$ .

Tap for more steps...

Step 4.1

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

$f (0) = \sqrt{(0) ((0) + 1)} - (0)$

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Multiply $0$ by $0 + 1$ .

$f (0) = \sqrt{0} - (0)$

Step 4.2.2

Rewrite $0$ as $0^{2}$ .

$f (0) = \sqrt{0^{2}} - (0)$

Step 4.2.3

Pull terms out from under the radical, assuming positive real numbers.

$f (0) = 0 - (0)$

Step 4.2.4

Subtract $0$ from $0$ .

$f (0) = 0$

Step 4.2.5

The final answer is $0$ .

$0$

Step 4.3

The $y$ value at $x = 0$ is $0$ .

$y = 0$

Step 5

Find the $y$ value at $x = 1$ .

Tap for more steps...

Step 5.1

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

$f (1) = \sqrt{(1) ((1) + 1)} - (1)$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1

Multiply $1 + 1$ by $1$ .

$f (1) = \sqrt{1 + 1} - (1)$

Step 5.2.1.2

Add $1$ and $1$ .

$f (1) = \sqrt{2} - (1)$

Step 5.2.1.3

Multiply $- 1$ by $1$ .

$f (1) = \sqrt{2} - 1$

Step 5.2.2

The final answer is $\sqrt{2} - 1$ .

$\sqrt{2} - 1$

Step 5.3

The $y$ value at $x = 1$ is $\sqrt{2} - 1$ .

$y = \sqrt{2} - 1$

Step 6

Find the $y$ value at $x = 2$ .

Tap for more steps...

Step 6.1

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

$f (2) = \sqrt{(2) ((2) + 1)} - (2)$

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Simplify each term.

Tap for more steps...

Step 6.2.1.1

Add $2$ and $1$ .

$f (2) = \sqrt{2 \cdot 3} - (2)$

Step 6.2.1.2

Multiply $2$ by $3$ .

$f (2) = \sqrt{6} - (2)$

Step 6.2.1.3

Multiply $- 1$ by $2$ .

$f (2) = \sqrt{6} - 2$

Step 6.2.2

The final answer is $\sqrt{6} - 2$ .

$\sqrt{6} - 2$

Step 6.3

The $y$ value at $x = 2$ is $\sqrt{6} - 2$ .

$y = \sqrt{6} - 2$

Step 7

List the points to graph.

$(- 1, 1), (- 2, 3.41421356), (- 3, 5.44948974), (0, 0), (1, 0.41421356), (2, 0.44948974)$

Step 8

Select a few points to graph.

$\begin{array}{c} x & y \\ - 3 & 5.449 \\ - 2 & 3.414 \\ - 1 & 1 \\ 0 & 0 \\ 1 & 0.414 \\ 2 & 0.449 \end{array}$

Step 9