Sure, let's explore the function [tex]\( y = -2 \log_2(x-1) + 3 \)[/tex] step-by-step. Here’s how you can find key points to plot and then sketch the graph:
1. Understand the Function Structure:
- The base function is [tex]\(\log_2(x-1)\)[/tex], which is defined for [tex]\(x > 1\)[/tex].
- The transformation includes multiplying by [tex]\(-2\)[/tex] and adding 3.
2. Determine Key Points:
- Select [tex]\(x\)[/tex] values greater than 1 such that [tex]\(\log_2(x-1)\)[/tex] results in simple values.
3. Evaluate at Selected Points:
- [tex]\(x = 2\)[/tex]:
[tex]\[
y = -2 \log_2(2-1) + 3 = -2 \log_2(1) + 3 = -2 \times 0 + 3 = 3
\][/tex]
Point: [tex]\((2, 3)\)[/tex]
- [tex]\(x = 3\)[/tex]:
[tex]\[
y = -2 \log_2(3-1) + 3 = -2 \log_2(2) + 3 = -2 \times 1 + 3 = 1
\][/tex]
Point: [tex]\((3, 1)\)[/tex]
- [tex]\(x = 5\)[/tex]:
[tex]\[
y = -2 \log_2(5-1) + 3 = -2 \log_2(4) + 3 = -2 \times 2 + 3 = -4 + 3 = -1
\][/tex]
Point: [tex]\((5, -1)\)[/tex]
- [tex]\(x = 9\)[/tex]:
[tex]\[
y = -2 \log_2(9-1) + 3 = -2 \log_2(8) + 3 = -2 \times 3 + 3 = -6 + 3 = -3
\][/tex]
Point: [tex]\((9, -3)\)[/tex]
4. Plot these points on a coordinate plane:
- [tex]\((2, 3)\)[/tex]
- [tex]\((3, 1)\)[/tex]
- [tex]\((5, -1)\)[/tex]
- [tex]\((9, -3)\)[/tex]
5. Graph the Curve:
- The overall shape of the curve decreases and is a logarithmic decay, since the coefficient of the logarithm function is negative.
- Plotting each of these points and drawing a smooth curve through them will help you sketch the graph accurately:
- At [tex]\(x=2\)[/tex], the curve should pass through [tex]\(y=3\)[/tex].
- At [tex]\(x=3\)[/tex], it passes through [tex]\(y=1\)[/tex].
- At [tex]\(x=5\)[/tex], it hits [tex]\(y=-1\)[/tex].
- At [tex]\(x=9\)[/tex], it reaches [tex]\(y=-3\)[/tex].
Following these calculations and plotting, you should have a graph resembling a downward logarithmic curve shifted up and flipped vertically. The general behavior near its horizontal asymptote will be moving towards [tex]\(y = 3\)[/tex] as [tex]\(x\)[/tex] approaches 1 from the right.
