Calculus Examples

$y = 2 x - \sqrt{x^{2} + 1}$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 2 x - \sqrt{x^{2} + 1}$

Step 1.2

Solve the equation.

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Step 1.2.1

Remove parentheses.

$0 = 2 x - \sqrt{x^{2} + 1}$

Step 1.2.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = \frac{\sqrt{3}}{3}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{\sqrt{3}}{3}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

Simplify $2 (0) - \sqrt{{(0)}^{2} + 1}$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Multiply $2$ by $0$ .

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

Step 2.2.3.1.2

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

$y = 0 - \sqrt{0 + 1}$

Step 2.2.3.1.3

Add $0$ and $1$ .

$y = 0 - \sqrt{1}$

Step 2.2.3.1.4

Any root of $1$ is $1$ .

$y = 0 - 1 \cdot 1$

Step 2.2.3.1.5

Multiply $- 1$ by $1$ .

$y = 0 - 1$

Step 2.2.3.2

Subtract $1$ from $0$ .

$y = - 1$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 1)$

Step 3

List the intersections.

x-intercept(s): $(\frac{\sqrt{3}}{3}, 0)$

y-intercept(s): $(0, - 1)$

Step 4