Precalculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Rewrite.

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

Apply the distributive property.

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

Step 2.3.2

Apply the distributive property.

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

Step 2.3.3

Apply the distributive property.

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

Step 2.4

Simplify and combine like terms.

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Step 2.4.1

Simplify each term.

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Step 2.4.1.1

Multiply $x$ by $x$ .

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

Step 2.4.1.2

Move $- 1$ to the left of $x$ .

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

Step 2.4.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 2.4.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 2.4.1.5

Multiply $- 1$ by $- 1$ .

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

Step 2.4.2

Subtract $x$ from $- x$ .

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

Step 3

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

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Step 3.1

Add $1$ to both sides of the equation.

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

Step 3.2

Add $1$ and $1$ .

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

Step 4

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 5

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

Interval Notation:

$[1, \infty)$

Set-Builder Notation:

${y | y \geq 1}$

Step 6

Determine the domain and range.

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

Range: $[1, \infty), {y | y \geq 1}$

Step 7