Algebra Examples

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

Step 1

Identify the degree of the function.

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Step 1.1

Simplify and reorder the polynomial.

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

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

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

Step 1.1.1.2

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

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Step 1.1.1.2.1

Apply the distributive property.

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

Step 1.1.1.2.2

Apply the distributive property.

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

Step 1.1.1.2.3

Apply the distributive property.

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

Step 1.1.1.3

Simplify and combine like terms.

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Step 1.1.1.3.1

Simplify each term.

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Step 1.1.1.3.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.1.1.3.1.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.1.1.3.1.1.2

Add $2$ and $2$ .

$x^{4} + x^{2} \cdot 2 + 2 x^{2} + 2 \cdot 2 + 3$

Step 1.1.1.3.1.2

Move $2$ to the left of $x^{2}$ .

$x^{4} + 2 \cdot x^{2} + 2 x^{2} + 2 \cdot 2 + 3$

Step 1.1.1.3.1.3

Multiply $2$ by $2$ .

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

Step 1.1.1.3.2

Add $2 x^{2}$ and $2 x^{2}$ .

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

Step 1.1.2

Add $4$ and $3$ .

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

Step 1.2

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

$x^{4} \to 4$

$4 x^{2} \to 2$

$7 \to 0$

Step 1.3

The largest exponent is the degree of the polynomial.

$4$

Step 2

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

Even

Step 3

Identify the leading coefficient.

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Step 3.1

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

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

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

Step 3.1.1.2

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

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Step 3.1.1.2.1

Apply the distributive property.

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

Step 3.1.1.2.2

Apply the distributive property.

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

Step 3.1.1.2.3

Apply the distributive property.

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

Step 3.1.1.3

Simplify and combine like terms.

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Step 3.1.1.3.1

Simplify each term.

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Step 3.1.1.3.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 3.1.1.3.1.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.1.1.3.1.1.2

Add $2$ and $2$ .

$x^{4} + x^{2} \cdot 2 + 2 x^{2} + 2 \cdot 2 + 3$

Step 3.1.1.3.1.2

Move $2$ to the left of $x^{2}$ .

$x^{4} + 2 \cdot x^{2} + 2 x^{2} + 2 \cdot 2 + 3$

Step 3.1.1.3.1.3

Multiply $2$ by $2$ .

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

Step 3.1.1.3.2

Add $2 x^{2}$ and $2 x^{2}$ .

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

Step 3.1.2

Add $4$ and $3$ .

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

Step 3.2

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

$x^{4}$

Step 3.3

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

$1$

Step 4

Since the leading coefficient is positive, the graph rises to the right.

Positive

Step 5

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 6

Determine the behavior.

Rises to the left and rises to the right

Step 7