Question

b. the function, ( g(x) ), is graphed to the right. the function, ( g(x) ), has already been vertically stretched by 3, and translated. write a new function if ( g(x) ) is translated 6 units left and 3 units up.

1. the function, ( g(x) ), is transformed such that the new function is ( h(x) ). identify the transformations to ( h(x) ).( g(x) = \frac{1}{2}(x + 4)^2 + 7 )( h(x) = 2(x)^2 - 5 )

1. write the transformed equation of ( y = \frac{1}{x} ), that has been shifted down 4 units, reflected across the ( x )-axis, & vertically stretched by a factor of 5.

1. the function ( y = 3|x| - 4 ) is transformed into a new function. it is moved to the right 2 units and up 4 units. what is the new function?

1. write the transformed equation of ( y = x^3 ), that has been shifted left 4 units, up 1 unit, reflected across the ( x )-axis, and vertically compressed by a factor of ( \frac{1}{4} ).

Explanation:

Step1: Identify base function from graph

The graph matches $g(x) = 2^x$ (passes through $(-1, 0.5), (0,1), (1,2)$).

Step2: Apply vertical stretch by 3

$g_1(x) = 3 \cdot 2^x$

Step3: Translate left 6 units

Replace $x$ with $x+6$: $g_2(x) = 3 \cdot 2^{(x+6)}$

Step4: Translate up 3 units

Add 3 to the function: $g_3(x) = 3 \cdot 2^{(x+6)} + 3$

Step1: Identify base transformation rules

For $g(x) = a(x-h)^3 + k$: $a$=vertical stretch/compression, $h$=horizontal shift, $k$=vertical shift.

Step2: Analyze vertical stretch factor

$a = \frac{1}{2}$: vertical compression by $\frac{1}{2}$

Step3: Analyze horizontal translation

$x+4 = x - (-4)$: 4 units left

Step4: Analyze vertical translation

$+7$: 7 units up

Step1: Apply x-axis reflection

Multiply by $-1$: $y = -\frac{1}{x}$

Step2: Apply vertical stretch by 5

Multiply by 5: $y = -\frac{5}{x}$

Step3: Apply vertical shift down 4

Subtract 4: $y = -\frac{5}{x} - 4$

Step1: Apply horizontal shift right 2

Replace $x$ with $x-2$: $y = 3|x-2| - 4$

Step2: Apply vertical shift up 4

Add 4: $y = 3|x-2| - 4 + 4$

Step3: Simplify the function

$y = 3|x-2|$

Answer:

$3 \cdot 2^{x+6} + 3$

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