Find the x and y Intercepts y=(-x-1)^2+3

$y = {(- x - 1)}^{2} + 3$

Step 1

Find the x-intercepts.

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

$0 = {(- x - 1)}^{2} + 3$

Solve the equation.

Rewrite the equation as ${(- x - 1)}^{2} + 3 = 0$ .

${(- x - 1)}^{2} + 3 = 0$

Subtract $3$ from both sides of the equation.

${(- x - 1)}^{2} = - 3$

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

$- x - 1 = \pm \sqrt{- 3}$

Simplify $\pm \sqrt{- 3}$ .

Rewrite $- 3$ as $- 1 (3)$ .

$- x - 1 = \pm \sqrt{- 1 (3)}$

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$- x - 1 = \pm \sqrt{- 1} \cdot \sqrt{3}$

Rewrite $\sqrt{- 1}$ as $i$ .

$- x - 1 = \pm i \sqrt{3}$

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

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

$- x - 1 = i \sqrt{3}$

Add $1$ to both sides of the equation.

$- x = i \sqrt{3} + 1$

Divide each term in $- x = i \sqrt{3} + 1$ by $- 1$ and simplify.

Divide each term in $- x = i \sqrt{3} + 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{i \sqrt{3}}{- 1} + \frac{1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{i \sqrt{3}}{- 1} + \frac{1}{- 1}$

Divide $x$ by $1$ .

$x = \frac{i \sqrt{3}}{- 1} + \frac{1}{- 1}$

Simplify the right side.

Simplify each term.

Move the negative one from the denominator of $\frac{i \sqrt{3}}{- 1}$ .

$x = - 1 \cdot (i \sqrt{3}) + \frac{1}{- 1}$

Rewrite $- 1 \cdot (i \sqrt{3})$ as $- (i \sqrt{3})$ .

$x = - (i \sqrt{3}) + \frac{1}{- 1}$

Divide $1$ by $- 1$ .

$x = - i \sqrt{3} - 1$

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

$- x - 1 = - i \sqrt{3}$

Add $1$ to both sides of the equation.

$- x = - i \sqrt{3} + 1$

Divide each term in $- x = - i \sqrt{3} + 1$ by $- 1$ and simplify.

Divide each term in $- x = - i \sqrt{3} + 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- i \sqrt{3}}{- 1} + \frac{1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- i \sqrt{3}}{- 1} + \frac{1}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- i \sqrt{3}}{- 1} + \frac{1}{- 1}$

Simplify the right side.

Simplify each term.

Dividing two negative values results in a positive value.

$x = \frac{i \sqrt{3}}{1} + \frac{1}{- 1}$

Divide $i \sqrt{3}$ by $1$ .

$x = i \sqrt{3} + \frac{1}{- 1}$

Divide $1$ by $- 1$ .

$x = i \sqrt{3} - 1$

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

$x = - i \sqrt{3} - 1, i \sqrt{3} - 1$

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

x-intercept(s): $None$

Step 2

Find the y-intercepts.

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

$y = {(- (0) - 1)}^{2} + 3$

Solve the equation.

Remove parentheses.

$y = {(- (0) - 1)}^{2} + 3$

Simplify ${(- (0) - 1)}^{2} + 3$ .

Simplify each term.

Multiply $- 1$ by $0$ .

$y = {(0 - 1)}^{2} + 3$

Subtract $1$ from $0$ .

$y = {(- 1)}^{2} + 3$

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

$y = 1 + 3$

Add $1$ and $3$ .

$y = 4$

y-intercept(s) in point form.

y-intercept(s): $(0, 4)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 4)$

Step 4

Find the x and y Intercepts y=(-x-1)^2+3

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