Question

identify the transformations of the graph of the parent function that produce the graph of the given function ( g(x) ). identify the asymptote of the graph of ( g(x) ).

1. ( g(x) = 3log(x + 4) + 1 )
2. ( g(x) = -\frac{1}{2}ln x + 4 )
3. ( g(x) = \frac{1}{10}log_6(x - 3) )
4. ( g(x) = -8log(x + \frac{1}{2}) - 1 )

identify the coordinates of the reference points on the graph of the parent function and their images on the graph of the given function ( g(x) ).

1. ( g(x) = -log(x + 3) - 2 )
2. ( g(x) = 5log_2(x - 1) - 4 )

identify the transformations of the graph of the parent function that produce the graph of the given function ( g(x) ). graph each function ( g(x) ) along with the graph of its parent function. include the reference points.

1. ( g(x) = -3log(x - 3) + 1 )
2. ( g(x) = 2log_{\frac{1}{2}}(x + 1) + 3 )

Explanation:

Problem 2: $g(x)=-\frac{1}{2}\ln x + 4$

Step1: Identify reflection

Reflection over the $x$-axis (due to $-$ sign)

Step2: Identify vertical shrink

Vertical shrink by factor $\frac{1}{2}$

Step3: Identify vertical shift

Shift up 4 units

Step4: Find vertical asymptote

Parent $\ln x$ has asymptote $x=0$; no horizontal shift, so asymptote $x=0$

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Problem 3: $g(x)=\frac{1}{10}\log_3(x-3)$

Step1: Identify vertical shrink

Vertical shrink by factor $\frac{1}{10}$

Step2: Identify horizontal shift

Shift right 3 units

Step3: Find vertical asymptote

Set $x-3=0$, so $x=3$

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Problem 5: $g(x)=-\log(x+3)-2$ (parent $f(x)=\log x$)

Step1: List parent reference points

Parent points: $(1,0), (10,1)$

Step2: Apply transformations to $(1,0)$

Reflect over $x$-axis: $(1,0)\to(1,0)$; shift left 3: $(1-3,0)=(-2,0)$; shift down 2: $(-2,0-2)=(-2,-2)$

Step3: Apply transformations to $(10,1)$

Reflect over $x$-axis: $(10,1)\to(10,-1)$; shift left 3: $(10-3,-1)=(7,-1)$; shift down 2: $(7,-1-2)=(7,-3)$

Step4: Find vertical asymptote

Set $x+3=0$, so $x=-3$

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Problem 6: $g(x)=5\log_2(x-1)-4$ (parent $f(x)=\log_2 x$)

Step1: List parent reference points

Parent points: $(1,0), (2,1), (4,2)$

Step2: Apply transformations to $(1,0)$

Vertical stretch by 5: $(1,0\times5)=(1,0)$; shift right 1: $(1+1,0)=(2,0)$; shift down 4: $(2,0-4)=(2,-4)$

Step3: Apply transformations to $(2,1)$

Vertical stretch by 5: $(2,1\times5)=(2,5)$; shift right 1: $(2+1,5)=(3,5)$; shift down 4: $(3,5-4)=(3,1)$

Step4: Apply transformations to $(4,2)$

Vertical stretch by 5: $(4,2\times5)=(4,10)$; shift right 1: $(4+1,10)=(5,10)$; shift down 4: $(5,10-4)=(5,6)$

Step5: Find vertical asymptote

Set $x-1=0$, so $x=1$

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Problem 8: $g(x)=2\log_3(x+1)+3$ (parent $f(x)=\log_3 x$)

Step1: Identify transformations

Vertical stretch by 2; shift left 1 unit; shift up 3 units

Step2: List parent reference points

Parent points: $(1,0), (3,1), (9,2)$

Step3: Transform $(1,0)$

Stretch: $(1,0\times2)=(1,0)$; shift left 1: $(1-1,0)=(0,0)$; shift up 3: $(0,0+3)=(0,3)$

Step4: Transform $(3,1)$

Stretch: $(3,1\times2)=(3,2)$; shift left 1: $(3-1,2)=(2,2)$; shift up 3: $(2,2+3)=(2,5)$

Step5: Transform $(9,2)$

Stretch: $(9,2\times2)=(9,4)$; shift left 1: $(9-1,4)=(8,4)$; shift up 3: $(8,4+3)=(8,7)$

Step6: Find vertical asymptote

Set $x+1=0$, so $x=-1$

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Answer:

Problem 2:

Transformations: Reflection over $x$-axis, vertical shrink by $\frac{1}{2}$, shift up 4 units. Asymptote: $x=0$

Problem 3:

Transformations: Vertical shrink by $\frac{1}{10}$, shift right 3 units. Asymptote: $x=3$

Problem 5:

Parent points: $(1,0), (10,1)$; Image points: $(-2,-2), (7,-3)$. Asymptote: $x=-3$

Problem 6:

Parent points: $(1,0), (2,1), (4,2)$; Image points: $(2,-4), (3,1), (5,6)$. Asymptote: $x=1$

Problem 8:

Transformations: Vertical stretch by 2, shift left 1 unit, shift up 3 units. Parent points: $(1,0), (3,1), (9,2)$; Image points: $(0,3), (2,5), (8,7)$. Asymptote: $x=-1$