Answer :
To sketch the graph of the function [tex]\( f(x) = -2 \log_{0.5}(2-x) + 1 \)[/tex] and its parent function [tex]\( g(x) = \log_{0.5}(2-x) \)[/tex], we need to follow several steps and transformations. Let's break these down.
### Step-by-Step Solution:
Step 1: Identify and understand the parent function [tex]\( g(x) \)[/tex]
The parent function here is [tex]\( g(x) = \log_{0.5}(2-x) \)[/tex]. This is a logarithmic function with base 0.5, which means it has a vertical asymptote at [tex]\( x = 2 \)[/tex] and it decreases as [tex]\( x \)[/tex] increases.
Step 2: Transformation of the parent function.
We'll go step-by-step through the transformation to obtain [tex]\( f(x) \)[/tex]:
1. Horizontal Shift:
- There is an inside shift by [tex]\((-x)\)[/tex], indicating a change.
- Here the shift affects the argument of the logarithm. The standard [tex]\( \log_{0.5}(2-x) \)[/tex] shifts [tex]\( x \)[/tex] such that it alters the starting point.
2. Vertical Stretch and Reflection:
- The coefficient [tex]\(-2\)[/tex] in [tex]\(-2 \log_{0.5}(2-x)\)[/tex] implies a vertical stretch by a factor of 2 and a reflection across the x-axis.
3. Vertical Shift:
- Adding 1 at the end ([tex]\(-2 \log_{0.5}(2-x) + 1\)[/tex]) shifts the entire graph up by 1 unit.
Step 3: Calculate 4 accuracy points
To plot the graph correctly, we calculate a few specific points.
Evaluate [tex]\( f(x) \)[/tex] at selected values of [tex]\( x \)[/tex]:
1. [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -2 \log_{0.5}(2-0) + 1 = -2 \log_{0.5}(2) + 1 = 3 \][/tex]
2. [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = -2 \log_{0.5}(1) + 1 = -2(0) + 1 = 1 \][/tex]
3. [tex]\( x = 1.5 \)[/tex]:
[tex]\[ f(1.5) = -2 \log_{0.5}(2-1.5) + 1 = -2 \log_{0.5}(0.5) + 1 = -2(-1) + 1 = 3 \][/tex]
4. [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = -2 \log_{0.5}(2-2) + 1 \][/tex]
Since [tex]\(\log_{0.5}(0)\)[/tex] is undefined (resulting in infinity), the function tends to [tex]\(-\infty\)[/tex].
[tex]\[ f(2) = -2 \log_{0.5}(0) + 1 \rightarrow -\infty \][/tex]
Step 4: Plot the Graph
Let's list the points computed:
- [tex]\( (0, 3) \)[/tex]
- [tex]\( (1, 1) \)[/tex]
- [tex]\( (1.5, -1) \)[/tex]
- [tex]\( (2, -\infty) \)[/tex]
### Graph Sketch
To sketch the graph:
1. Start by plotting points [tex]\( (0, 3) \)[/tex], [tex]\( (1, 1) \)[/tex], and [tex]\( (1.5, -1) \)[/tex].
2. Draw the vertical asymptote line [tex]\( x = 2 \)[/tex].
3. Identify that the parent function [tex]\( g(x) = \log_{0.5}(2-x) \)[/tex] decreases and plots properly.
Plot points and transformations accurately:
- For [tex]\( (0, 3) \)[/tex], mark the point and place the vertical line for visibility.
- Point [tex]\( (1, 1) \)[/tex] clearly aligns with 1.
- Point [tex]\( (1.5, -1) \)[/tex].
Graph experiences sharp drop closer towards [tex]\( x = 2 \)[/tex].
### Note:
- Use a strong scale on both axes, keeping [tex]\( -\infty \)[/tex].
- Logarithmic functions experience gradual and rapid changes near vertical asymptotes.
- Illustrate transformed function from the reflection in parent function to final positional changes.
This way, you comprehensively address transformations, accuracy points and sketch the transformed function of [tex]\( f(x) \)[/tex].
### Step-by-Step Solution:
Step 1: Identify and understand the parent function [tex]\( g(x) \)[/tex]
The parent function here is [tex]\( g(x) = \log_{0.5}(2-x) \)[/tex]. This is a logarithmic function with base 0.5, which means it has a vertical asymptote at [tex]\( x = 2 \)[/tex] and it decreases as [tex]\( x \)[/tex] increases.
Step 2: Transformation of the parent function.
We'll go step-by-step through the transformation to obtain [tex]\( f(x) \)[/tex]:
1. Horizontal Shift:
- There is an inside shift by [tex]\((-x)\)[/tex], indicating a change.
- Here the shift affects the argument of the logarithm. The standard [tex]\( \log_{0.5}(2-x) \)[/tex] shifts [tex]\( x \)[/tex] such that it alters the starting point.
2. Vertical Stretch and Reflection:
- The coefficient [tex]\(-2\)[/tex] in [tex]\(-2 \log_{0.5}(2-x)\)[/tex] implies a vertical stretch by a factor of 2 and a reflection across the x-axis.
3. Vertical Shift:
- Adding 1 at the end ([tex]\(-2 \log_{0.5}(2-x) + 1\)[/tex]) shifts the entire graph up by 1 unit.
Step 3: Calculate 4 accuracy points
To plot the graph correctly, we calculate a few specific points.
Evaluate [tex]\( f(x) \)[/tex] at selected values of [tex]\( x \)[/tex]:
1. [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -2 \log_{0.5}(2-0) + 1 = -2 \log_{0.5}(2) + 1 = 3 \][/tex]
2. [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = -2 \log_{0.5}(1) + 1 = -2(0) + 1 = 1 \][/tex]
3. [tex]\( x = 1.5 \)[/tex]:
[tex]\[ f(1.5) = -2 \log_{0.5}(2-1.5) + 1 = -2 \log_{0.5}(0.5) + 1 = -2(-1) + 1 = 3 \][/tex]
4. [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = -2 \log_{0.5}(2-2) + 1 \][/tex]
Since [tex]\(\log_{0.5}(0)\)[/tex] is undefined (resulting in infinity), the function tends to [tex]\(-\infty\)[/tex].
[tex]\[ f(2) = -2 \log_{0.5}(0) + 1 \rightarrow -\infty \][/tex]
Step 4: Plot the Graph
Let's list the points computed:
- [tex]\( (0, 3) \)[/tex]
- [tex]\( (1, 1) \)[/tex]
- [tex]\( (1.5, -1) \)[/tex]
- [tex]\( (2, -\infty) \)[/tex]
### Graph Sketch
To sketch the graph:
1. Start by plotting points [tex]\( (0, 3) \)[/tex], [tex]\( (1, 1) \)[/tex], and [tex]\( (1.5, -1) \)[/tex].
2. Draw the vertical asymptote line [tex]\( x = 2 \)[/tex].
3. Identify that the parent function [tex]\( g(x) = \log_{0.5}(2-x) \)[/tex] decreases and plots properly.
Plot points and transformations accurately:
- For [tex]\( (0, 3) \)[/tex], mark the point and place the vertical line for visibility.
- Point [tex]\( (1, 1) \)[/tex] clearly aligns with 1.
- Point [tex]\( (1.5, -1) \)[/tex].
Graph experiences sharp drop closer towards [tex]\( x = 2 \)[/tex].
### Note:
- Use a strong scale on both axes, keeping [tex]\( -\infty \)[/tex].
- Logarithmic functions experience gradual and rapid changes near vertical asymptotes.
- Illustrate transformed function from the reflection in parent function to final positional changes.
This way, you comprehensively address transformations, accuracy points and sketch the transformed function of [tex]\( f(x) \)[/tex].
