Question

in exercises 15–22, describe the transformation from the graph of ( f ) to the graph of ( g ). (see example 3.)

15.

( x )	( -2 )	( -1 )	( 0 )	( 1 )
( f(x) )	( -9 )	( -2 )	( -1 )	( 0 )
( g(x) = f(x) + k )	( -11 )	( -4 )	( -3 )	( -2 )

16.

( x )	( -1 )	( 0 )	( 1 )	( 2 )
( f(x) )	( -2 )	( 0 )	( -2 )	( 16 )
( g(x) = a cdot f(x) )	( 2 )	( 0 )	( 2 )	( -16 )

1. ( g(x) = f(x) + k ) (with a graph of two parabolas, ( f ) and ( g ) on a coordinate grid)

1. ( g(x) = -f(x - h) ) (with a graph of two curves, ( f ) and ( g ) on a coordinate grid)

19.

( x )	( -2 )	( -1 )	( 0 )	( 1 )
( f(x) )	( -85 )	( -2 )	( -5 )	( -4 )
( g(x) )	( 89 )	( 6 )	( 9 )	( 8 )

20.

( x )	( -1 )	( 0 )	( 1 )	( 2 )
( f(x) )	( 8 )	( 3 )	( 0 )	( -13 )
( g(x) )	( 26 )	( 11 )	( 2 )	( -37 )

1. (with a graph of two curves, ( f ) (black) and ( g ) (blue) on a coordinate grid)

1. (with a graph of two parabolas, ( f ) (black) and ( g ) (blue) on a coordinate grid)

Explanation:

Step1: Solve for k (Problem 15)

$k = g(x) - f(x)$
For $x=-2$: $k = -11 - (-9) = -2$
Verify with $x=-1$: $k = -4 - (-2) = -2$

Step2: Describe transformation (15)

Vertical shift down 2 units.

Step3: Solve for a (Problem 16)

$a = \frac{g(x)}{f(x)}$
For $x=-1$: $a = \frac{2}{-2} = -1$
Verify with $x=2$: $a = \frac{-16}{16} = -1$

Step4: Describe transformation (16)

Reflection over x-axis.

Step5: Identify k (Problem 17)

From graph, g is f shifted up 3.
$k=3$, vertical shift up 3 units.

Step6: Identify h (Problem 18)

g is $-f(x-h)$: reflected over x-axis, shifted right 1 unit.

Step7: Find g(x) form (Problem 19)

$g(x) = -f(x) + 4$: reflection over x-axis, vertical shift up 4 units.
Check $x=-2$: $-(-85)+4=89$, matches g(x).

Step8: Find g(x) form (Problem 20)

$g(x) = 3f(x) + 2$: vertical stretch by 3, shift up 2.
Check $x=-1$: $3(8)+2=26$, matches g(x).

Step9: Describe transformation (Problem 21)

g is f shifted right 4, up 3 units.

Step10: Describe transformation (Problem 22)

g is f shifted left 0, vertical shift up 2 units (or vertical stretch + shift, graph shows g is f shifted up 2, same shape).

Answer:

1. Vertical shift down 2 units
2. Reflection across the x-axis
3. Vertical shift up 3 units
4. Reflection across the x-axis and horizontal shift right 1 unit
5. Reflection across the x-axis and vertical shift up 4 units
6. Vertical stretch by a factor of 3 and vertical shift up 2 units
7. Horizontal shift right 4 units and vertical shift up 3 units
8. Vertical shift up 2 units