Question

the function f(x) is represented by the given table. what are the corresponding values of the given g(x)?

1. the function f(x) is represented by the table below. what are the corresponding values of g(x) for the transformation g(x) = 6f(x)?

x	f(x)
-7	8
-3	3
0	-1
2	7
10	5

a. | x | g(x) |

-7	30
-3	42
0	0
2	18
10	48

b. | x | g(x) |

-7	-42
-3	-18
0	6
2	12
10	60

c. | x | g(x) |

-7	-48
-3	-18
0	6
2	-42
10	-30

d. | x | g(x) |

-7	48
-3	18
0	-6
2	42
10	30

1. write an equation for the following transformation of y = x: a vertical compression by a factor of \\(\frac{1}{4}\\)

a. \\(y = -4x\\)
b. \\(y = -\frac{1}{4}x\\)
c. \\(y = 4x\\)
d. \\(y = \frac{1}{4}x\\)
what transformations change the graph of f(x) to the graph of g(x)?

1. \\(f(x) = x^2\\); \\(g(x) = (x + 5)^2 - 9\\)

a. the graph of g(x) is the graph of f(x) translated to the left 5 units and down 9 units.
b. the graph of g(x) is the graph of f(x) translated to the up 5 units and right 9 units.
c. the graph of g(x) is the graph of f(x) translated to the down 5 units and left 9 units.
d. the graph of g(x) is the graph of f(x) translated to the right 5 units and up 9 units.

Explanation:

Question 4

Step1: Recall the transformation rule

For the transformation \( g(x) = 6f(x) \), we need to multiply each \( f(x) \) value by 6.

Step2: Calculate \( g(x) \) for each \( x \)

• When \( x = -7 \), \( f(x)=8 \), so \( g(-7)=6\times8 = 48 \)
• When \( x = -3 \), \( f(x)=3 \), so \( g(-3)=6\times3 = 18 \)
• When \( x = 0 \), \( f(x)=-1 \), so \( g(0)=6\times(-1)= - 6 \)
• When \( x = 2 \), \( f(x)=7 \), so \( g(2)=6\times7 = 42 \)
• When \( x = 10 \), \( f(x)=5 \), so \( g(10)=6\times5 = 30 \)

Comparing with the options, option d has these values.

Question 5

Step1: Recall vertical compression rule

A vertical compression of a function \( y = f(x) \) by a factor of \( a \) (where \( 0 < a<1 \)) is given by \( y=af(x) \).

Step2: Apply the rule to \( y = x \)

For a vertical compression by a factor of \( \frac{1}{4} \), we multiply the function \( y = x \) by \( \frac{1}{4} \). So the equation becomes \( y=\frac{1}{4}x \), which is option d.

Question 6

Step1: Recall horizontal and vertical translation rules

• For horizontal translation: If we have \( f(x + h) \), the graph of \( f(x) \) is translated left by \( h \) units (if \( h>0 \)) and right by \( h \) units (if \( h < 0 \)).
• For vertical translation: If we have \( f(x)+k \), the graph of \( f(x) \) is translated up by \( k \) units (if \( k>0 \)) and down by \( k \) units (if \( k < 0 \)).

Step2: Analyze \( g(x)=(x + 5)^{2}-9 \) compared to \( f(x)=x^{2} \)

• For the horizontal part: \( x \) is replaced by \( x + 5 \), so \( h = 5>0 \), which means a translation left by 5 units.
• For the vertical part: We have \( - 9 \) added to the function, so \( k=-9<0 \), which means a translation down by 9 units.

So the transformation is a translation left 5 units and down 9 units, which is option a.

Answer:

d.

x	g(x)
-3	18
0	-6
2	42
10	30