Answer:
10. A translation of 4 units down.
11. A translation of 6 units left.
12. A reflection in the y-axis, followed by a vertical compression by a factor of ¹/₂.
13. A horizontal compression by a factor of ¹/₅, followed by a reflection in the x-axis.
14. A translation of 4 units left, followed by a vertical stretch by a factor of 3, followed by a translation of 6 units down.
15. A horizontal compression by a factor of ²/₃, followed by a reflection in the x-axis, followed by a translation of 5 units down.
Step-by-step explanation:
Transformations
[tex]\textsf{For $a > 0$}:[/tex]
[tex]f(x+a) \implies f(x) \: \textsf{translated $a$ units left}.[/tex]
[tex]f(x-a) \implies f(x) \: \textsf{translated $a$ units right}.[/tex]
[tex]f(x)+a \implies f(x) \: \textsf{translated $a$ units up}.[/tex]
[tex]f(x)-a \implies f(x) \: \textsf{translated $a$ units down}.[/tex]
[tex]a\:f(x) \implies f(x) \: \textsf{stretched parallel to the $y$-axis (vertically) by a factor of $a$}.[/tex]
[tex]f(ax) \implies f(x) \: \textsf{stretched parallel to the $x$-axis (horizontally) by a factor of $\dfrac{1}{a}$}.[/tex]
[tex]-f(x) \implies f(x) \: \textsf{reflected in the $x$-axis}.[/tex]
[tex]f(-x) \implies f(x) \: \textsf{reflected in the $y$-axis}.[/tex]
Question 10
[tex]\textsf{Given}: \quad g(x)=f(x)-4[/tex]
Therefore, the transformation to get from f(x) to g(x) is:
- Translation of 4 units down.
Question 11
[tex]\textsf{Given}: \quad g(x)=f(x+6)[/tex]
Therefore, the transformation to get from f(x) to g(x) is:
- Translation of 6 units left.
Question 12
[tex]\textsf{Given}: \quad g(x)=\dfrac{1}{2}f(-x)[/tex]
Therefore, the series of transformations to get from f(x) to g(x) is:
- Reflection in the y-axis.
- Vertical compression by a factor of ¹/₂.
Question 13
[tex]\textsf{Given}: \quad g(x)=-f(5x)[/tex]
Therefore, the series of transformations to get from f(x) to g(x) is:
- Horizontal compression by a factor of ¹/₅.
- Reflection in the x-axis.
Question 14
[tex]\textsf{Given}: \quad g(x)=3f(x+4)-6[/tex]
Therefore, the series of transformations to get from f(x) to g(x) is:
- Translation of 4 units left.
- Vertical stretch by a factor of 3.
- Translation of 6 units down.
Question 15
[tex]\textsf{Given}: \quad g(x)=-f\left(\dfrac{3}{2}x\right)-5[/tex]
Therefore, the series of transformations to get from f(x) to g(x) is:
- Horizontal compression by a factor of ²/₃.
- Reflection in the x-axis.
- Translation of 5 units down.
