text: # 005 Sample Final A, Question 17
  
  (diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
  
  Question Graph the following function,
  
  ${\displaystyle f(x)=\log _{2}(x+1)+2}$
  
  Make sure to label any asymptotes, and at least two points on the graph.
  
  Foundations
  1) What is the basic graph of ${\displaystyle f(x)=\log _{2}(x+1)+2}$?
  2) How is the graph ${\displaystyle g(x)=x^{3}+2}$ obtained from ${\displaystyle f(x)=x^{3}}$?
  3) How is the graph ${\displaystyle g(x)=(x+1)^{2}}$ obtained from ${\displaystyle f(x)=x^{2}}$?
  1) The basic graph is ${\displaystyle y=\log _{2}(x)}$.
  2) The graph of ${\displaystyle g(x)}$ is obtained by shifting the graph of ${\displaystyle f(x)}$ up 2 unit.
  3) The graph of ${\displaystyle g(x)}$ is obtained by shifting the graph of ${\displaystyle f(x)}$ to the left by 1 unit.
  
  Solution:
  
  Step 1:
  We start with the basic graph of ${\displaystyle g(x)=\log _{2}(x)}$.
  To get the graph of ${\displaystyle f(x)}$ from ${\displaystyle g(x)}$, we shift the graph of ${\displaystyle g(x)}$ up 2 and to the left by 1.
  Step 2:
  Two points on the graph are (0, 2) and (1, 3). There is a vertical asymptote at ${\displaystyle x=-1}$.
  Two points on the graph are (0, 2) and (1, 3). There is a vertical asymptote at ${\displaystyle x=-1}$.
explanation: The document contains a math question asking to graph the function \( f(x) = \log_2(x+1) + 2 \) and label any asymptotes and at least two points on the graph. The document also includes a solution with step-by-step instructions and the final answer.
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distilabel_metadata.role: system
distilabel_metadata.content: # 005 Sample Final A, Question 17
  
  (diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
  
  Question Graph the following function,
  
  ${\displaystyle f(x)=\log _{2}(x+1)+2}$
  
  Make sure to label any asymptotes, and at least two points on the graph.
  
  Foundations
  1) What is the basic graph of ${\displaystyle f(x)=\log _{2}(x+1)+2}$?
  2) How is the graph ${\displaystyle g(x)=x^{3}+2}$ obtained from ${\displaystyle f(x)=x^{3}}$?
  3) How is the graph ${\displaystyle g(x)=(x+1)^{2}}$ obtained from ${\displaystyle f(x)=x^{2}}$?
  1) The basic graph is ${\displaystyle y=\log _{2}(x)}$.
  2) The graph of ${\displaystyle g(x)}$ is obtained by shifting the graph of ${\displaystyle f(x)}$ up 2 unit.
  3) The graph of ${\displaystyle g(x)}$ is obtained by shifting the graph of ${\displaystyle f(x)}$ to the left by 1 unit.
  
  Solution:
  
  Step 1:
  We start with the basic graph of ${\displaystyle g(x)=\log _{2}(x)}$.
  To get the graph of ${\displaystyle f(x)}$ from ${\displaystyle g(x)}$, we shift the graph of ${\displaystyle g(x)}$ up 2 and to the left by 1.
  Step 2:
  Two points on the graph are (0, 2) and (1, 3). There is a vertical asymptote at ${\displaystyle x=-1}$.
  Two points on the graph are (0, 2) and (1, 3). There is a vertical asymptote at ${\displaystyle x=-1}$.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question asking to graph the function \( f(x) = \log_2(x+1) + 2 \) and label any asymptotes and at least two points on the graph. The document also includes a solution with step-by-step instructions and the final answer.
  
  Label: question-and-answer