Algebra Examples

$f (x) = \frac{3}{2} (x - 2)$

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

Apply the distributive property.

$\frac{3}{2} x + \frac{3}{2} \cdot - 2$

Step 1.1.2

Combine $\frac{3}{2}$ and $x$ .

$\frac{3 x}{2} + \frac{3}{2} \cdot - 2$

Step 1.1.3

Cancel the common factor of $2$ .

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Step 1.1.3.1

Factor $2$ out of $- 2$ .

$\frac{3 x}{2} + \frac{3}{2} \cdot (2 (- 1))$

Step 1.1.3.2

Cancel the common factor.

$\frac{3 x}{2} + \frac{3}{2} \cdot (2 \cdot - 1)$

Step 1.1.3.3

Rewrite the expression.

$\frac{3 x}{2} + 3 \cdot - 1$

Step 1.1.4

Multiply $3$ by $- 1$ .

$\frac{3 x}{2} - 3$

Step 1.2

The largest exponent is the degree of the polynomial.

$1$

Step 2

Since the degree is odd, the ends of the function will point in the opposite directions.

Odd

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

Apply the distributive property.

$\frac{3}{2} x + \frac{3}{2} \cdot - 2$

Step 3.1.2

Combine $\frac{3}{2}$ and $x$ .

$\frac{3 x}{2} + \frac{3}{2} \cdot - 2$

Step 3.1.3

Cancel the common factor of $2$ .

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Step 3.1.3.1

Factor $2$ out of $- 2$ .

$\frac{3 x}{2} + \frac{3}{2} \cdot (2 (- 1))$

Step 3.1.3.2

Cancel the common factor.

$\frac{3 x}{2} + \frac{3}{2} \cdot (2 \cdot - 1)$

Step 3.1.3.3

Rewrite the expression.

$\frac{3 x}{2} + 3 \cdot - 1$

Step 3.1.4

Multiply $3$ by $- 1$ .

$\frac{3 x}{2} - 3$

Step 3.2

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

$\frac{3 x}{2}$

Step 3.3

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

$\frac{3}{2}$

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.

Falls to the left and rises to the right

Step 7