Calculus Examples

$y = 4 \log_{3} (x + 1) + 1$

Step 1

Find the asymptotes.

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Step 1.1

Find where the expression $\log_{3} ({(x + 1)}^{4}) + 1$ is undefined.

$x = - 1$

Step 1.2

Ignoring the logarithm, consider the rational function $R (x) = \frac{a x^{n}}{b x^{m}}$ where $n$ is the degree of the numerator and $m$ is the degree of the denominator.

1. If $n < m$ , then the x-axis, $y = 0$ , is the horizontal asymptote.

2. If $n = m$ , then the horizontal asymptote is the line $y = \frac{a}{b}$ .

3. If $n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 1.3

There are no horizontal asymptotes because $Q (x)$ is $1$ .

No Horizontal Asymptotes

Step 1.4

No oblique asymptotes are present for logarithmic and trigonometric functions.

No Oblique Asymptotes

Step 1.5

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 1$

No Horizontal Asymptotes

Vertical Asymptotes: $x = - 1$

No Horizontal Asymptotes

Step 2

Find the point at $x = 0$ .

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Step 2.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = 4 \log_{3} ((0) + 1) + 1$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Add $0$ and $1$ .

$f (0) = 4 \log_{3} (1) + 1$

Step 2.2.1.2

Logarithm base $3$ of $1$ is $0$ .

$f (0) = 4 \cdot 0 + 1$

Step 2.2.1.3

Multiply $4$ by $0$ .

$f (0) = 0 + 1$

Step 2.2.2

Add $0$ and $1$ .

$f (0) = 1$

Step 2.2.3

The final answer is $1$ .

$1$

Step 2.3

Convert $1$ to decimal.

$y = 1$

Step 3

Find the point at $x = 2$ .

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Step 3.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = 4 \log_{3} ((2) + 1) + 1$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Add $2$ and $1$ .

$f (2) = 4 \log_{3} (3) + 1$

Step 3.2.1.2

Logarithm base $3$ of $3$ is $1$ .

$f (2) = 4 \cdot 1 + 1$

Step 3.2.1.3

Multiply $4$ by $1$ .

$f (2) = 4 + 1$

Step 3.2.2

Add $4$ and $1$ .

$f (2) = 5$

Step 3.2.3

The final answer is $5$ .

$5$

Step 3.3

Convert $5$ to decimal.

$y = 5$

Step 4

Find the point at $x = 8$ .

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Step 4.1

Replace the variable $x$ with $8$ in the expression.

$f (8) = 4 \log_{3} ((8) + 1) + 1$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Add $8$ and $1$ .

$f (8) = 4 \log_{3} (9) + 1$

Step 4.2.1.2

Logarithm base $3$ of $9$ is $2$ .

$f (8) = 4 \cdot 2 + 1$

Step 4.2.1.3

Multiply $4$ by $2$ .

$f (8) = 8 + 1$

Step 4.2.2

Add $8$ and $1$ .

$f (8) = 9$

Step 4.2.3

The final answer is $9$ .

$9$

Step 4.3

Convert $9$ to decimal.

$y = 9$

Step 5

The log function can be graphed using the vertical asymptote at $x = - 1$ and the points $(0, 1), (2, 5), (8, 9)$ .

Vertical Asymptote: $x = - 1$

$\begin{array}{c} x & y \\ 0 & 1 \\ 2 & 5 \\ 8 & 9 \end{array}$

Step 6