Algebra Examples

$y = - {(x + 3)}^{2} - 6$

Step 1

Simplify $- {(x + 3)}^{2} - 6$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$y = - ((x + 3) (x + 3)) - 6$

Step 1.1.2

Expand $(x + 3) (x + 3)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$y = - (x (x + 3) + 3 (x + 3)) - 6$

Step 1.1.2.2

Apply the distributive property.

$y = - (x \cdot x + x \cdot 3 + 3 (x + 3)) - 6$

Step 1.1.2.3

Apply the distributive property.

$y = - (x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3) - 6$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $x$ by $x$ .

$y = - (x^{2} + x \cdot 3 + 3 x + 3 \cdot 3) - 6$

Step 1.1.3.1.2

Move $3$ to the left of $x$ .

$y = - (x^{2} + 3 \cdot x + 3 x + 3 \cdot 3) - 6$

Step 1.1.3.1.3

Multiply $3$ by $3$ .

$y = - (x^{2} + 3 x + 3 x + 9) - 6$

Step 1.1.3.2

Add $3 x$ and $3 x$ .

$y = - (x^{2} + 6 x + 9) - 6$

Step 1.1.4

Apply the distributive property.

$y = - x^{2} - (6 x) - 1 \cdot 9 - 6$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Multiply $6$ by $- 1$ .

$y = - x^{2} - 6 x - 1 \cdot 9 - 6$

Step 1.1.5.2

Multiply $- 1$ by $9$ .

$y = - x^{2} - 6 x - 9 - 6$

Step 1.2

Subtract $6$ from $- 9$ .

$y = - x^{2} - 6 x - 15$

Step 2

Identify the degree of the function.

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Step 2.1

Identify the exponents on the variables in each term, and add them together to find the degree of each term.

$- x^{2} \to 2$

$- 6 x \to 1$

$- 15 \to 0$

Step 2.2

The largest exponent is the degree of the polynomial.

$2$

Step 3

Since the degree is even, the ends of the function will point in the same direction.

Even

Step 4

Identify the leading coefficient.

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Step 4.1

The leading term in a polynomial is the term with the highest degree.

$- x^{2}$

Step 4.2

The leading coefficient in a polynomial is the coefficient of the leading term.

$- 1$

Step 5

Since the leading coefficient is negative, the graph falls to the right.

Negative

Step 6

Use the degree of the function, as well as the sign of the leading coefficient to determine the behavior.

1. Even and Positive: Rises to the left and rises to the right.

2. Even and Negative: Falls to the left and falls to the right.

3. Odd and Positive: Falls to the left and rises to the right.

4. Odd and Negative: Rises to the left and falls to the right

Step 7

Determine the behavior.

Falls to the left and falls to the right

Step 8