Answer :
Let’s break this down step by step according to the transformations applied to the function [tex]\( f(x) = 3x - 7 \)[/tex]:
1. The function [tex]\( f \)[/tex] translated 6 units up and 2 units right:
- Translating [tex]\( f(x) \)[/tex] 2 units to the right means we have to replace [tex]\( x \)[/tex] by [tex]\( x-2 \)[/tex].
- Translating [tex]\( f(x) \)[/tex] 6 units up means we add 6 to the function.
- Hence, the transformed function is:
[tex]\[ g(x) = f(x - 2) + 6 \][/tex]
Calculating this:
[tex]\[ f(x-2) = 3(x-2) - 7 = 3x - 6 - 7 = 3x - 13 \][/tex]
Then adding 6:
[tex]\[ g(x) = 3x - 13 + 6 = 3x - 7 \][/tex]
So, the equivalent function rule is [tex]\( g(x) = 3x - 7 \)[/tex].
2. The function [tex]\( f \)[/tex] reflected about the y-axis and translated 7 units left:
- Reflecting about the y-axis means replacing [tex]\( x \)[/tex] by [tex]\( -x \)[/tex].
- Translating 7 units to the left means replacing [tex]\( x \)[/tex] by [tex]\( x+7 \)[/tex].
- Hence, the transformed function is:
[tex]\[ g(x) = f(-x + 7) \][/tex]
Calculating this:
[tex]\[ f(-x + 7) = 3(-x + 7) - 7 = -3x + 21 - 7 = -3x + 14 \][/tex]
So, the equivalent function rule is [tex]\( g(x) = -3x + 14 \)[/tex].
3. The function [tex]\( f \)[/tex] stretched vertically by a factor of 4 and translated up by 9 units:
- Stretching vertically by a factor of 4 means multiplying the entire function by 4.
- Translating the function up by 9 units means adding 9.
- Hence, the transformed function is:
[tex]\[ g(x) = 4 f(x) + 9 \][/tex]
Calculating this:
[tex]\[ 4 f(x) = 4(3x - 7) = 12x - 28 \][/tex]
Then adding 9:
[tex]\[ g(x) = 12x - 28 + 9 = 12x - 19 \][/tex]
So, the equivalent function rule is [tex]\( g(x) = 12x - 19 \)[/tex].
4. The function [tex]\( f \)[/tex] stretched vertically by a factor of 2 and translated up by 5 units:
- Stretching vertically by a factor of 2 means multiplying the entire function by 2.
- Translating the function up by 5 units means adding 5.
- Hence, the transformed function is:
[tex]\[ g(x) = 2 f(x) + 5 \][/tex]
Calculating this:
[tex]\[ 2 f(x) = 2(3x - 7) = 6x - 14 \][/tex]
Then adding 5:
[tex]\[ g(x) = 6x - 14 + 5 = 6x - 9 \][/tex]
So, the equivalent function rule is [tex]\( g(x) = 6x - 9 \)[/tex].
To summarize the matches:
- The function [tex]\( f \)[/tex] translated 6 units up and 2 units right: [tex]\( g(x) = 3x - 7 \)[/tex]
- The function [tex]\( f \)[/tex] reflected about the y-axis and translated 7 units left: [tex]\( g(x) = -3x + 14 \)[/tex]
- The function [tex]\( f \)[/tex] stretched vertically by a factor of 4 and translated up by 9 units: [tex]\( g(x) = 12x - 19 \)[/tex]
- The function [tex]\( f \)[/tex] stretched vertically by a factor of 2 and translated up by 5 units: [tex]\( g(x) = 6x - 9 \)[/tex]
1. The function [tex]\( f \)[/tex] translated 6 units up and 2 units right:
- Translating [tex]\( f(x) \)[/tex] 2 units to the right means we have to replace [tex]\( x \)[/tex] by [tex]\( x-2 \)[/tex].
- Translating [tex]\( f(x) \)[/tex] 6 units up means we add 6 to the function.
- Hence, the transformed function is:
[tex]\[ g(x) = f(x - 2) + 6 \][/tex]
Calculating this:
[tex]\[ f(x-2) = 3(x-2) - 7 = 3x - 6 - 7 = 3x - 13 \][/tex]
Then adding 6:
[tex]\[ g(x) = 3x - 13 + 6 = 3x - 7 \][/tex]
So, the equivalent function rule is [tex]\( g(x) = 3x - 7 \)[/tex].
2. The function [tex]\( f \)[/tex] reflected about the y-axis and translated 7 units left:
- Reflecting about the y-axis means replacing [tex]\( x \)[/tex] by [tex]\( -x \)[/tex].
- Translating 7 units to the left means replacing [tex]\( x \)[/tex] by [tex]\( x+7 \)[/tex].
- Hence, the transformed function is:
[tex]\[ g(x) = f(-x + 7) \][/tex]
Calculating this:
[tex]\[ f(-x + 7) = 3(-x + 7) - 7 = -3x + 21 - 7 = -3x + 14 \][/tex]
So, the equivalent function rule is [tex]\( g(x) = -3x + 14 \)[/tex].
3. The function [tex]\( f \)[/tex] stretched vertically by a factor of 4 and translated up by 9 units:
- Stretching vertically by a factor of 4 means multiplying the entire function by 4.
- Translating the function up by 9 units means adding 9.
- Hence, the transformed function is:
[tex]\[ g(x) = 4 f(x) + 9 \][/tex]
Calculating this:
[tex]\[ 4 f(x) = 4(3x - 7) = 12x - 28 \][/tex]
Then adding 9:
[tex]\[ g(x) = 12x - 28 + 9 = 12x - 19 \][/tex]
So, the equivalent function rule is [tex]\( g(x) = 12x - 19 \)[/tex].
4. The function [tex]\( f \)[/tex] stretched vertically by a factor of 2 and translated up by 5 units:
- Stretching vertically by a factor of 2 means multiplying the entire function by 2.
- Translating the function up by 5 units means adding 5.
- Hence, the transformed function is:
[tex]\[ g(x) = 2 f(x) + 5 \][/tex]
Calculating this:
[tex]\[ 2 f(x) = 2(3x - 7) = 6x - 14 \][/tex]
Then adding 5:
[tex]\[ g(x) = 6x - 14 + 5 = 6x - 9 \][/tex]
So, the equivalent function rule is [tex]\( g(x) = 6x - 9 \)[/tex].
To summarize the matches:
- The function [tex]\( f \)[/tex] translated 6 units up and 2 units right: [tex]\( g(x) = 3x - 7 \)[/tex]
- The function [tex]\( f \)[/tex] reflected about the y-axis and translated 7 units left: [tex]\( g(x) = -3x + 14 \)[/tex]
- The function [tex]\( f \)[/tex] stretched vertically by a factor of 4 and translated up by 9 units: [tex]\( g(x) = 12x - 19 \)[/tex]
- The function [tex]\( f \)[/tex] stretched vertically by a factor of 2 and translated up by 5 units: [tex]\( g(x) = 6x - 9 \)[/tex]
