Algebra Examples

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

Step 1

The parent function is the simplest form of the type of function given.

$y = x^{2}$

Step 2

Simplify ${(x + 2)}^{2} + 3$ .

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

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

$y = (x + 2) (x + 2) + 3$

Step 2.1.2

Expand $(x + 2) (x + 2)$ using the FOIL Method.

Tap for more steps...

Step 2.1.2.1

Apply the distributive property.

$y = x (x + 2) + 2 (x + 2) + 3$

Step 2.1.2.2

Apply the distributive property.

$y = x \cdot x + x \cdot 2 + 2 (x + 2) + 3$

Step 2.1.2.3

Apply the distributive property.

$y = x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2 + 3$

Step 2.1.3

Simplify and combine like terms.

Tap for more steps...

Step 2.1.3.1

Simplify each term.

Tap for more steps...

Step 2.1.3.1.1

Multiply $x$ by $x$ .

$y = x^{2} + x \cdot 2 + 2 x + 2 \cdot 2 + 3$

Step 2.1.3.1.2

Move $2$ to the left of $x$ .

$y = x^{2} + 2 \cdot x + 2 x + 2 \cdot 2 + 3$

Step 2.1.3.1.3

Multiply $2$ by $2$ .

$y = x^{2} + 2 x + 2 x + 4 + 3$

Step 2.1.3.2

Add $2 x$ and $2 x$ .

$y = x^{2} + 4 x + 4 + 3$

Step 2.2

Add $4$ and $3$ .

$y = x^{2} + 4 x + 7$

Step 3

Assume that $y = x^{2}$ is $f (x) = x^{2}$ and $y = {(x + 2)}^{2} + 3$ is $g (x) = x^{2} + 4 x + 7$ .

$f (x) = x^{2}$

$g (x) = x^{2} + 4 x + 7$

Step 4

The transformation being described is from $f (x) = x^{2}$ to $g (x) = x^{2} + 4 x + 7$ .

$f (x) = x^{2} \to g (x) = x^{2} + 4 x + 7$

Step 5

Find the vertex form of $g (x) = x^{2} + 4 x + 7$ .

Tap for more steps...

Step 5.1

Complete the square for $x^{2} + 4 x + 7$ .

Tap for more steps...

Step 5.1.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1$

$b = 4$

$c = 7$

Step 5.1.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 5.1.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Tap for more steps...

Step 5.1.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{4}{2 \cdot 1}$

Step 5.1.3.2

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 5.1.3.2.1

Factor $2$ out of $4$ .

$d = \frac{2 \cdot 2}{2 \cdot 1}$

Step 5.1.3.2.2

Cancel the common factors.

Tap for more steps...

Step 5.1.3.2.2.1

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot 2}{2 (1)}$

Step 5.1.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 2}{2 \cdot 1}$

Step 5.1.3.2.2.3

Rewrite the expression.

$d = \frac{2}{1}$

Step 5.1.3.2.2.4

Divide $2$ by $1$ .

$d = 2$

Step 5.1.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Tap for more steps...

Step 5.1.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 7 - \frac{4^{2}}{4 \cdot 1}$

Step 5.1.4.2

Simplify the right side.

Tap for more steps...

Step 5.1.4.2.1

Simplify each term.

Tap for more steps...

Step 5.1.4.2.1.1

Cancel the common factor of $4^{2}$ and $4$ .

Tap for more steps...

Step 5.1.4.2.1.1.1

Factor $4$ out of $4^{2}$ .

$e = 7 - \frac{4 \cdot 4}{4 \cdot 1}$

Step 5.1.4.2.1.1.2

Cancel the common factors.

Tap for more steps...

Step 5.1.4.2.1.1.2.1

Factor $4$ out of $4 \cdot 1$ .

$e = 7 - \frac{4 \cdot 4}{4 (1)}$

Step 5.1.4.2.1.1.2.2

Cancel the common factor.

$e = 7 - \frac{4 \cdot 4}{4 \cdot 1}$

Step 5.1.4.2.1.1.2.3

Rewrite the expression.

$e = 7 - \frac{4}{1}$

Step 5.1.4.2.1.1.2.4

Divide $4$ by $1$ .

$e = 7 - 1 \cdot 4$

Step 5.1.4.2.1.2

Multiply $- 1$ by $4$ .

$e = 7 - 4$

Step 5.1.4.2.2

Subtract $4$ from $7$ .

$e = 3$

Step 5.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + 2)}^{2} + 3$ .

${(x + 2)}^{2} + 3$

Step 5.2

Set $y$ equal to the new right side.

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

Step 6

The horizontal shift depends on the value of $h$ . The horizontal shift is described as:

$g (x) = f (x + h)$ - The graph is shifted to the left $h$ units.

$g (x) = f (x - h)$ - The graph is shifted to the right $h$ units.

Horizontal Shift: Left $2$ Units

Step 7

The vertical shift depends on the value of $k$ . The vertical shift is described as:

$g (x) = f (x) + k$ - The graph is shifted up $k$ units.

$g (x) = f (x) - k$ - The graph is shifted down $k$ units.

Vertical Shift: Up $3$ Units

Step 8

The graph is reflected about the x-axis when $g (x) = - f (x)$ .

Reflection about the x-axis: None

Step 9

The graph is reflected about the y-axis when $g (x) = f (- x)$ .

Reflection about the y-axis: None

Step 10

Compressing and stretching depends on the value of $a$ .

When $a$ is greater than $1$ : Vertically stretched

When $a$ is between $0$ and $1$ : Vertically compressed

Vertical Compression or Stretch: None

Step 11

Compare and list the transformations.

Parent Function: $y = x^{2}$

Horizontal Shift: Left $2$ Units

Vertical Shift: Up $3$ Units

Reflection about the x-axis: None

Reflection about the y-axis: None

Vertical Compression or Stretch: None

Step 12