Math / Algebra

QuestionIn Exercises 41-46, describe how to transform the graph of $y=\ln x$ into the graph of the given function. Sketch the graph by hand and support your sketch with a grapher.
41. $f(x)=\ln (x+3)$
42. $f(x)=\ln (x)+2$
43. $f(x)=\ln (-x)+3$
44. $f(x)=\ln (-x)-2$
45. $f(x)=\ln (2-x)$
46. $f(x)=\ln (5-x)$

Studdy Solution

STEP 1

1. The base function is $y = \ln x$ .
2. Transformations include horizontal shifts, vertical shifts, and reflections.
3. We will describe transformations for each function from Exercises 41-46.

STEP 2

1. Identify horizontal transformations.
2. Identify vertical transformations.
3. Identify reflections.
4. Sketch the transformed graph.

STEP 3

For $f(x) = \ln(x+3)$ : - This is a horizontal shift. - The graph of $y = \ln x$ is shifted 3 units to the left.

STEP 4

For $f(x) = \ln(x) + 2$ : - This is a vertical shift. - The graph of $y = \ln x$ is shifted 2 units upward.

STEP 5

For $f(x) = \ln(-x) + 3$ : - This involves a reflection and a vertical shift. - Reflect the graph of $y = \ln x$ across the y-axis. - Then shift the graph 3 units upward.

STEP 6

For $f(x) = \ln(-x) - 2$ : - This involves a reflection and a vertical shift. - Reflect the graph of $y = \ln x$ across the y-axis. - Then shift the graph 2 units downward.

STEP 7

For $f(x) = \ln(2-x)$ : - This involves a reflection and a horizontal shift. - Reflect the graph of $y = \ln x$ across the y-axis. - Then shift the graph 2 units to the right.

STEP 8

For $f(x) = \ln(5-x)$ : - This involves a reflection and a horizontal shift. - Reflect the graph of $y = \ln x$ across the y-axis. - Then shift the graph 5 units to the right.

STEP 9

Sketch each transformed graph based on the described transformations. - Use a grapher to verify the accuracy of each sketch.

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Studdy Solution

STEP 1

1. The base function is $y = \ln x$ .
2. Transformations include horizontal shifts, vertical shifts, and reflections.
3. We will describe transformations for each function from Exercises 41-46.

STEP 2

1. Identify horizontal transformations.
2. Identify vertical transformations.
3. Identify reflections.
4. Sketch the transformed graph.

STEP 3

For $f(x) = \ln(x+3)$ : - This is a horizontal shift. - The graph of $y = \ln x$ is shifted 3 units to the left.

STEP 4

For $f(x) = \ln(x) + 2$ : - This is a vertical shift. - The graph of $y = \ln x$ is shifted 2 units upward.

STEP 5

For $f(x) = \ln(-x) + 3$ : - This involves a reflection and a vertical shift. - Reflect the graph of $y = \ln x$ across the y-axis. - Then shift the graph 3 units upward.

STEP 6

For $f(x) = \ln(-x) - 2$ : - This involves a reflection and a vertical shift. - Reflect the graph of $y = \ln x$ across the y-axis. - Then shift the graph 2 units downward.

STEP 7

For $f(x) = \ln(2-x)$ : - This involves a reflection and a horizontal shift. - Reflect the graph of $y = \ln x$ across the y-axis. - Then shift the graph 2 units to the right.

STEP 8

For $f(x) = \ln(5-x)$ : - This involves a reflection and a horizontal shift. - Reflect the graph of $y = \ln x$ across the y-axis. - Then shift the graph 5 units to the right.

STEP 9

Sketch each transformed graph based on the described transformations. - Use a grapher to verify the accuracy of each sketch.

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Studdy Solution

STEP 1

1. The base function is y=ln⁡x y = \ln x y=lnx.2. Transformations include horizontal shifts, vertical shifts, and reflections.3. We will describe transformations for each function from Exercises 41-46.

STEP 2

1. Identify horizontal transformations.2. Identify vertical transformations.3. Identify reflections.4. Sketch the transformed graph.

STEP 3

For f(x)=ln⁡(x+3) f(x) = \ln(x+3) f(x)=ln(x+3): - This is a horizontal shift. - The graph of y=ln⁡x y = \ln x y=lnx is shifted 3 units to the left.

STEP 4

For f(x)=ln⁡(x)+2 f(x) = \ln(x) + 2 f(x)=ln(x)+2: - This is a vertical shift. - The graph of y=ln⁡x y = \ln x y=lnx is shifted 2 units upward.

STEP 5

For f(x)=ln⁡(−x)+3 f(x) = \ln(-x) + 3 f(x)=ln(−x)+3: - This involves a reflection and a vertical shift. - Reflect the graph of y=ln⁡x y = \ln x y=lnx across the y-axis. - Then shift the graph 3 units upward.

STEP 6

For f(x)=ln⁡(−x)−2 f(x) = \ln(-x) - 2 f(x)=ln(−x)−2: - This involves a reflection and a vertical shift. - Reflect the graph of y=ln⁡x y = \ln x y=lnx across the y-axis. - Then shift the graph 2 units downward.

STEP 7

For f(x)=ln⁡(2−x) f(x) = \ln(2-x) f(x)=ln(2−x): - This involves a reflection and a horizontal shift. - Reflect the graph of y=ln⁡x y = \ln x y=lnx across the y-axis. - Then shift the graph 2 units to the right.

STEP 8

For f(x)=ln⁡(5−x) f(x) = \ln(5-x) f(x)=ln(5−x): - This involves a reflection and a horizontal shift. - Reflect the graph of y=ln⁡x y = \ln x y=lnx across the y-axis. - Then shift the graph 5 units to the right.

STEP 9

Sketch each transformed graph based on the described transformations. - Use a grapher to verify the accuracy of each sketch.

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1. The base function is $y = \ln x$ .
2. Transformations include horizontal shifts, vertical shifts, and reflections.
3. We will describe transformations for each function from Exercises 41-46.

1. Identify horizontal transformations.
2. Identify vertical transformations.
3. Identify reflections.
4. Sketch the transformed graph.

For $f(x) = \ln(x+3)$ : - This is a horizontal shift. - The graph of $y = \ln x$ is shifted 3 units to the left.

For $f(x) = \ln(x) + 2$ : - This is a vertical shift. - The graph of $y = \ln x$ is shifted 2 units upward.

For $f(x) = \ln(-x) + 3$ : - This involves a reflection and a vertical shift. - Reflect the graph of $y = \ln x$ across the y-axis. - Then shift the graph 3 units upward.

For $f(x) = \ln(-x) - 2$ : - This involves a reflection and a vertical shift. - Reflect the graph of $y = \ln x$ across the y-axis. - Then shift the graph 2 units downward.

For $f(x) = \ln(2-x)$ : - This involves a reflection and a horizontal shift. - Reflect the graph of $y = \ln x$ across the y-axis. - Then shift the graph 2 units to the right.

For $f(x) = \ln(5-x)$ : - This involves a reflection and a horizontal shift. - Reflect the graph of $y = \ln x$ across the y-axis. - Then shift the graph 5 units to the right.