Find the Domain and Range y-6=(x+1)^2

$y - 6 = {(x + 1)}^{2}$

Step 1

Simplify ${(x + 1)}^{2}$ .

Rewrite.

$y - 6 = 0 + 0 + {(x + 1)}^{2}$

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$y - 6 = (x + 1) (x + 1)$

Expand $(x + 1) (x + 1)$ using the FOIL Method.

Apply the distributive property.

$y - 6 = x (x + 1) + 1 (x + 1)$

Apply the distributive property.

$y - 6 = x \cdot x + x \cdot 1 + 1 (x + 1)$

Apply the distributive property.

$y - 6 = x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y - 6 = x^{2} + x \cdot 1 + 1 x + 1 \cdot 1$

Multiply $x$ by $1$ .

$y - 6 = x^{2} + x + 1 x + 1 \cdot 1$

Multiply $x$ by $1$ .

$y - 6 = x^{2} + x + x + 1 \cdot 1$

Multiply $1$ by $1$ .

$y - 6 = x^{2} + x + x + 1$

Add $x$ and $x$ .

$y - 6 = x^{2} + 2 x + 1$

Step 2

Move all terms not containing $y$ to the right side of the equation.

Add $6$ to both sides of the equation.

$y = x^{2} + 2 x + 1 + 6$

Add $1$ and $6$ .

$y = x^{2} + 2 x + 7$

Step 3

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:

${x | x \in ℝ}$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[6, \infty)$

Set-Builder Notation:

${y | y \geq 6}$

Step 5

Determine the domain and range.

Domain: $(- \infty, \infty), {x | x \in ℝ}$

Range: $[6, \infty), {y | y \geq 6}$

Step 6

Find the Domain and Range y-6=(x+1)^2

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