Question

2.11 logarithmic functions
ap precalculus
2.11 practice
sketch a graph of the transformation of $f(x) = \log_{5} x$ onto the graph. label each graph.

1. $g(x) = 3\log_{5}(x + 2) - 4$
2. $h(x) = 3\log_{5}(3 - x) + 1$
3. $j(x) = -\log_{5}(x - 3) - 2$

find all relevant information from the given function. sketch a graph. no calculator.

1. $f(x) = \ln(x - 3) + 5$

asymptote:
domain: range:
end behavior:
graph:

1. $f(x) = -2\log_{2}(x + 3) - 6$

asymptote:
domain: range:
end behavior:
graph:
the algebros from flippedmath.com

Explanation:

Step1: Analyze Asymptote

For \( f(x) = -2\log_2(x + 3) - 6 \), the vertical asymptote of a logarithmic function \( \log_b(u) \) is at \( u = 0 \). Here, \( u=x + 3 \), so set \( x+3=0 \), we get \( x=-3 \).

Step2: Determine Domain

The argument of the logarithm must be positive. So \( x + 3>0\), which gives \( x>-3 \). Thus, domain is \( (-3, \infty) \).

Step3: Find Range

Logarithmic functions \( \log_b(u) \) have a range of \( (-\infty, \infty) \). The transformations (vertical stretch by -2 and vertical shift down 6) do not change the range. So range is \( (-\infty, \infty) \).

Step4: Analyze End Behavior

As \( x
ightarrow - 3^+ \), \( x + 3
ightarrow0^+ \), \( \log_2(x + 3)
ightarrow-\infty \), then \( -2\log_2(x + 3)
ightarrow\infty \), and \( -2\log_2(x + 3)-6
ightarrow\infty \). As \( x
ightarrow\infty \), \( x + 3
ightarrow\infty \), \( \log_2(x + 3)
ightarrow\infty \), then \( -2\log_2(x + 3)
ightarrow-\infty \), and \( -2\log_2(x + 3)-6
ightarrow-\infty \).

Step5: Describe Graph Transformation

The parent function is \( y = \log_2x \). First, reflect over x - axis (because of -2), vertically stretch by factor 2, shift left 3 units (because of \( x+3 \)), and shift down 6 units (because of -6).

Answer:

• Asymptote: \( x = - 3 \)
• Domain: \( (-3, \infty) \)
• Range: \( (-\infty, \infty) \)
• End Behavior: As \( x

ightarrow - 3^+ \), \( f(x)
ightarrow\infty \); as \( x
ightarrow\infty \), \( f(x)
ightarrow-\infty \)

• Graph: Reflect \( y = \log_2x \) over x - axis, vertically stretch by 2, shift left 3, shift down 6.