Algebra Examples

$g (x) = 5 - \frac{7}{2} x - 3 x^{2}$

Step 1

Identify the degree of the function.

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $x$ and $\frac{7}{2}$ .

$5 - \frac{x \cdot 7}{2} - 3 x^{2}$

Step 1.1.2

Move $7$ to the left of $x$ .

$5 - \frac{7 x}{2} - 3 x^{2}$

Step 1.2

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

$5 \to 0$

$- \frac{7 x}{2} \to 1$

$- 3 x^{2} \to 2$

Step 1.3

The largest exponent is the degree of the polynomial.

$2$

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

Combine $x$ and $\frac{7}{2}$ .

$5 - \frac{x \cdot 7}{2} - 3 x^{2}$

Step 3.1.1.2

Move $7$ to the left of $x$ .

$5 - \frac{7 x}{2} - 3 x^{2}$

Step 3.1.2

Simplify the expression.

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Step 3.1.2.1

Move $5$ .

$- \frac{7 x}{2} - 3 x^{2} + 5$

Step 3.1.2.2

Reorder $- \frac{7 x}{2}$ and $- 3 x^{2}$ .

$- 3 x^{2} - \frac{7 x}{2} + 5$

Step 3.2

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

$- 3 x^{2}$

Step 3.3

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

$- 3$

Step 4

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

Negative

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 falls to the right

Step 7