To plot the graph of the function [tex]\( y = -\log(x - 2) + 3 \)[/tex], we need to follow a systematic approach. Let's break it down step-by-step:
1. Understanding the Function:
- The function involves a logarithmic term [tex]\(\log(x - 2)\)[/tex].
- The function is shifted horizontally by 2 units to the right due to the [tex]\(x - 2\)[/tex] inside the logarithm.
- The negative sign in front of the logarithm ([tex]\(-\log\)[/tex]) will reflect the graph across the x-axis.
- The "+3" outside of the logarithm shifts the graph vertically upwards by 3 units.
2. Domain of the Function:
- The logarithmic function [tex]\(\log(x - 2)\)[/tex] is defined when the argument is positive, i.e., [tex]\(x - 2 > 0\)[/tex].
- Therefore, the domain of the function is [tex]\( x > 2 \)[/tex].
3. Vertical Asymptote:
- As [tex]\( x \)[/tex] approaches 2 from the right ([tex]\( x \to 2^+ \)[/tex]), [tex]\( \log(x - 2) \to -\infty \)[/tex].
- Because of the negative sign, [tex]\(-\log(x - 2)\)[/tex] will tend to [tex]\(+\infty\)[/tex].
- Adding 3 to [tex]\(+\infty\)[/tex] means the function approaches [tex]\(+\infty\)[/tex].
- Therefore, there is a vertical asymptote at [tex]\( x = 2 \)[/tex].
4. Behavior as [tex]\( x \to \infty \)[/tex]:
- As [tex]\( x \)[/tex] increases, [tex]\( x - 2 \)[/tex] increases, thus [tex]\( \log(x - 2) \)[/tex] also increases.
- [tex]\(-\log(x - 2) \)[/tex] will become more negative.
- Adding 3 results in a large negative value.
- Therefore, as [tex]\( x \to \infty \)[/tex], [tex]\( y \to -\infty \)[/tex].
5. Key Points and Plotting:
- Let's find a few key points to help plot the graph:
- For [tex]\( x = 3 \)[/tex]:
[tex]\[
y = -\log(3 - 2) + 3 = -\log(1) + 3 = -0 + 3 = 3
\][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[
y = -\log(4 - 2) + 3 = -\log(2) + 3 \approx -0.693 + 3 \approx 2.307
\][/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[
y = -\log(5 - 2) + 3 = -\log(3) + 3 \approx -1.099 + 3 \approx 1.901
\][/tex]
6. Sketching the Graph:
- Start by drawing a vertical asymptote at [tex]\( x = 2 \)[/tex].
- Plot the key points [tex]\((3, 3)\)[/tex], [tex]\((4, 2.307)\)[/tex], and [tex]\((5, 1.901)\)[/tex].
- Connect these points smoothly, showing that the function rapidly increases as [tex]\( x \to 2 \)[/tex] from the right and gradually decreases as [tex]\( x \to \infty \)[/tex].
Here's a sketch of the graph:
[tex]\[
\begin{array}{c|c}
x & y\\
\hline
2^+ & +\infty \\
3 & 3 \\
4 & 2.307 \\
5 & 1.901 \\
\vdots & \downarrow -\infty
\end{array}
\][/tex]
The graph of [tex]\( y = -\log(x - 2) + 3 \)[/tex] is a curve that:
- Has a vertical asymptote at [tex]\( x = 2 \)[/tex].
- Passes through the point (3, 3).
- Decreases as [tex]\( x \)[/tex] increases, eventually heading towards negative infinity as [tex]\( x \)[/tex] grows larger.
This detailed examination of the function allows us to visualize and interpret the behavior of its graph.
