Trigonometry Examples

$f (x) = \log (x^{2} + 6 x - 16)$

Step 1

Find the asymptotes.

Tap for more steps...

Step 1.1

Set the argument of the logarithm equal to zero.

$x^{2} + 6 x - 16 = 0$

Step 1.2

Solve for $x$ .

Tap for more steps...

Step 1.2.1

Factor $x^{2} + 6 x - 16$ using the AC method.

Tap for more steps...

Step 1.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 16$ and whose sum is $6$ .

$- 2, 8$

Step 1.2.1.2

Write the factored form using these integers.

$(x - 2) (x + 8) = 0$

Step 1.2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 2 = 0$

$x + 8 = 0$

Step 1.2.3

Set $x - 2$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.3.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 1.2.3.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.4

Set $x + 8$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.4.1

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Step 1.2.4.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 1.2.5

The final solution is all the values that make $(x - 2) (x + 8) = 0$ true.

$x = 2, - 8$

Step 1.3

The vertical asymptote occurs at $x = 2, x = - 8$ .

Vertical Asymptote: $x = 2, x = - 8$

Step 2

Find the point at $x = - 11$ .

Tap for more steps...

Step 2.1

Replace the variable $x$ with $- 11$ in the expression.

$f (- 11) = \log ({(- 11)}^{2} + 6 (- 11) - 16)$

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Raise $- 11$ to the power of $2$ .

$f (- 11) = \log (121 + 6 (- 11) - 16)$

Step 2.2.1.2

Multiply $6$ by $- 11$ .

$f (- 11) = \log (121 - 66 - 16)$

Step 2.2.2

Simplify by subtracting numbers.

Tap for more steps...

Step 2.2.2.1

Subtract $66$ from $121$ .

$f (- 11) = \log (55 - 16)$

Step 2.2.2.2

Subtract $16$ from $55$ .

$f (- 11) = \log (39)$

Step 2.2.3

The final answer is $\log (39)$ .

$\log (39)$

Step 2.3

Convert $\log (39)$ to decimal.

$y = 1.5910646$

Step 3

Find the point at $x = - 10$ .

Tap for more steps...

Step 3.1

Replace the variable $x$ with $- 10$ in the expression.

$f (- 10) = \log ({(- 10)}^{2} + 6 (- 10) - 16)$

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Raise $- 10$ to the power of $2$ .

$f (- 10) = \log (100 + 6 (- 10) - 16)$

Step 3.2.1.2

Multiply $6$ by $- 10$ .

$f (- 10) = \log (100 - 60 - 16)$

Step 3.2.2

Simplify by subtracting numbers.

Tap for more steps...

Step 3.2.2.1

Subtract $60$ from $100$ .

$f (- 10) = \log (40 - 16)$

Step 3.2.2.2

Subtract $16$ from $40$ .

$f (- 10) = \log (24)$

Step 3.2.3

The final answer is $\log (24)$ .

$\log (24)$

Step 3.3

Convert $\log (24)$ to decimal.

$y = 1.38021124$

Step 4

Find the point at $x = - 9$ .

Tap for more steps...

Step 4.1

Replace the variable $x$ with $- 9$ in the expression.

$f (- 9) = \log ({(- 9)}^{2} + 6 (- 9) - 16)$

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

Raise $- 9$ to the power of $2$ .

$f (- 9) = \log (81 + 6 (- 9) - 16)$

Step 4.2.1.2

Multiply $6$ by $- 9$ .

$f (- 9) = \log (81 - 54 - 16)$

Step 4.2.2

Simplify by subtracting numbers.

Tap for more steps...

Step 4.2.2.1

Subtract $54$ from $81$ .

$f (- 9) = \log (27 - 16)$

Step 4.2.2.2

Subtract $16$ from $27$ .

$f (- 9) = \log (11)$

Step 4.2.3

The final answer is $\log (11)$ .

$\log (11)$

Step 4.3

Convert $\log (11)$ to decimal.

$y = 1.04139268$

Step 5

The log function can be graphed using the vertical asymptote at $x = 2, x = - 8$ and the points $(- 11, 1.5910646), (- 10, 1.38021124), (- 9, 1.04139268)$ .

Vertical Asymptote: $x = 2, x = - 8$

$\begin{array}{c} x & y \\ - 11 & 1.591 \\ - 10 & 1.38 \\ - 9 & 1.041 \end{array}$

Step 6