Worksheet for graphing logarithmic functions in Infinite Algebra 2.

Since I can't view or access images directly, I’ll help you solve the graphing logarithmic functions problem based on the text you provided. Here are the six logarithmic functions:

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General Form of a Logarithmic Function:

$$
y = \log_b(x - h) + k
$$

- Domain: $ x > h $ (since log is only defined for positive inputs)
- Range: All real numbers ($ (-\infty, \infty) $)
- Vertical asymptote: $ x = h $
- Transformations:
- $ h $: horizontal shift (right if $ h > 0 $, left if $ h < 0 $)
- $ k $: vertical shift (up if $ k > 0 $, down if $ k < 0 $)

We will analyze each function step by step.

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1) $ y = \log_6(x - 1) - 5 $

- Base: 6 (greater than 1 → increasing)
- Horizontal shift: $ x - 1 $ → shift right by 1 unit
- Vertical shift: $ -5 $ → shift down by 5 units
- Domain: $ x - 1 > 0 $ → $ x > 1 $
- Range: All real numbers
- Vertical asymptote: $ x = 1 $

Graph sketch:
- Start with parent graph $ y = \log_6(x) $
- Shift right 1 → $ y = \log_6(x - 1) $
- Shift down 5 → $ y = \log_6(x - 1) - 5 $
- Plot key points:
- $ x = 2 $: $ y = \log_6(1) - 5 = 0 - 5 = -5 $
- $ x = 7 $: $ y = \log_6(6) - 5 = 1 - 5 = -4 $
- $ x = 37 $: $ y = \log_6(36) - 5 = \log_6(6^2) - 5 = 2 - 5 = -3 $

Asymptote at $ x = 1 $, curve increases slowly from left to right.

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2) $ y = \log_3(x - 1) + 3 $

- Base: 3
- Horizontal shift: right 1
- Vertical shift: up 3
- Domain: $ x > 1 $
- Range: all real numbers
- Asymptote: $ x = 1 $

Key points:
- $ x = 2 $: $ y = \log_3(1) + 3 = 0 + 3 = 3 $
- $ x = 4 $: $ y = \log_3(3) + 3 = 1 + 3 = 4 $
- $ x = 10 $: $ y = \log_3(9) + 3 = 2 + 3 = 5 $

Plot these and draw curve approaching $ x = 1 $ from right.

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3) $ y = \log_4(x - 3) - 5 $

- Base: 4
- Horizontal shift: right 3
- Vertical shift: down 5
- Domain: $ x > 3 $
- Range: all real numbers
- Asymptote: $ x = 3 $

Key points:
- $ x = 4 $: $ y = \log_4(1) - 5 = 0 - 5 = -5 $
- $ x = 7 $: $ y = \log_4(4) - 5 = 1 - 5 = -4 $
- $ x = 19 $: $ y = \log_4(16) - 5 = 2 - 5 = -3 $

Sketch curve starting near $ x = 3 $, increasing slowly.

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4) $ y = \log_2(x - 1) + 3 $

- Base: 2
- Right 1, up 3
- Domain: $ x > 1 $
- Range: all real numbers
- Asymptote: $ x = 1 $

Key points:
- $ x = 2 $: $ y = \log_2(1) + 3 = 0 + 3 = 3 $
- $ x = 3 $: $ y = \log_2(2) + 3 = 1 + 3 = 4 $
- $ x = 5 $: $ y = \log_2(4) + 3 = 2 + 3 = 5 $

Draw curve approaching $ x = 1 $ from right.

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5) $ y = \log_4(x + 1) - 4 $

- Base: 4
- Horizontal shift: $ x + 1 $ → left 1
- Vertical shift: down 4
- Domain: $ x + 1 > 0 $ → $ x > -1 $
- Range: all real numbers
- Asymptote: $ x = -1 $

Key points:
- $ x = 0 $: $ y = \log_4(1) - 4 = 0 - 4 = -4 $
- $ x = 3 $: $ y = \log_4(4) - 4 = 1 - 4 = -3 $
- $ x = 15 $: $ y = \log_4(16) - 4 = 2 - 4 = -2 $

Curve starts near $ x = -1 $, increases slowly.

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6) $ y = \log_2(x + 1) + 1 $

- Base: 2
- Left 1, up 1
- Domain: $ x > -1 $
- Range: all real numbers
- Asymptote: $ x = -1 $

Key points:
- $ x = 0 $: $ y = \log_2(1) + 1 = 0 + 1 = 1 $
- $ x = 1 $: $ y = \log_2(2) + 1 = 1 + 1 = 2 $
- $ x = 3 $: $ y = \log_2(4) + 1 = 2 + 1 = 3 $

Sketch curve approaching $ x = -1 $ from right.

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Summary Table

| Problem | Equation | Domain | Range | Vertical Asymptote |
|--------|----------|--------|-------|---------------------|
| 1 | $ y = \log_6(x - 1) - 5 $ | $ x > 1 $ | $ (-\infty, \infty) $ | $ x = 1 $ |
| 2 | $ y = \log_3(x - 1) + 3 $ | $ x > 1 $ | $ (-\infty, \infty) $ | $ x = 1 $ |
| 3 | $ y = \log_4(x - 3) - 5 $ | $ x > 3 $ | $ (-\infty, \infty) $ | $ x = 3 $ |
| 4 | $ y = \log_2(x - 1) + 3 $ | $ x > 1 $ | $ (-\infty, \infty) $ | $ x = 1 $ |
| 5 | $ y = \log_4(x + 1) - 4 $ | $ x > -1 $ | $ (-\infty, \infty) $ | $ x = -1 $ |
| 6 | $ y = \log_2(x + 1) + 1 $ | $ x > -1 $ | $ (-\infty, \infty) $ | $ x = -1 $ |

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How to Sketch Each Graph:

1. Draw the vertical asymptote as a dashed line.
2. Plot at least two key points using the transformations.
3. Use the fact that logarithmic graphs increase slowly and pass through the point where the argument is 1 (i.e., $ \log_b(1) = 0 $).
4. Connect the points with a smooth curve approaching the asymptote.

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Let me know if you'd like me to generate a visual description or help you plot specific points!