Calculus Examples

$y = 2 x^{2} - 1$

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 $x$ .

$0 = 2 x^{2} - 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $2 x^{2} - 1 = 0$ .

$2 x^{2} - 1 = 0$

Step 1.2.2

Add $1$ to both sides of the equation.

$2 x^{2} = 1$

Step 1.2.3

Divide each term in $2 x^{2} = 1$ by $2$ and simplify.

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Step 1.2.3.1

Divide each term in $2 x^{2} = 1$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{1}{2}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{1}{2}$

Step 1.2.3.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{1}{2}$

Step 1.2.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{1}{2}}$

Step 1.2.5

Simplify $\pm \sqrt{\frac{1}{2}}$ .

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Step 1.2.5.1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{2}}$

Step 1.2.5.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{2}}$

Step 1.2.5.3

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 1.2.5.4

Combine and simplify the denominator.

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Step 1.2.5.4.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.2.5.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.2.5.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.2.5.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.2.5.4.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.2.5.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.2.5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.2.5.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.2.5.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.2.5.4.6.4

Cancel the common factor of $2$ .

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Step 1.2.5.4.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.2.5.4.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{2}}{2^{1}}$

Step 1.2.5.4.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{2}}{2}$

Step 1.2.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.6.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 1.2.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{2}}{2}$

Step 1.2.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{\sqrt{2}}{2}, 0), (- \frac{\sqrt{2}}{2}, 0)$

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 2 \cdot 0^{2} - 1$

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

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

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

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

$y = 2 \cdot 0 - 1$

Step 2.2.3.1.2

Multiply $2$ by $0$ .

$y = 0 - 1$

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): $(\frac{\sqrt{2}}{2}, 0), (- \frac{\sqrt{2}}{2}, 0)$

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

Step 4