Sample quiz on combined transformations
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1. If the transformation $2f(x)+3$ of a parent function $f(x)$ is written as $af[k(x-d)]+c$, then:
   
   $a=2,k=1,d=1,c=3$
   $a=2,k=0,d=0,c=3$
   $a=2,k=1,d=0,c=3$
   $a=2,k=0,d=1,c=3$.
   
2. If the transformation $-3f(x-5)+3$ of a parent function $f(x)$ is written as $af[k(x-d)]+c$, then:
   
   $a=-3,k=0,d=5,c=3$
   $a=-3,k=1,d=5,c=3$
   $a=-3,k=1,d=-5,c=3$
   $a=3,k=1,d=5,c=3$
3. If the transformation $5f(x+2)-7$ of a parent function $f(x)$ is written as $af[k(x-d)]+c$, then:
   
   $a=5,k=1,d=2,c=-7$
   $a=5,k=-1,d=2,c=-7$
   $a=5,k=1,d=-2,c=-7$
   $a=5,k=1,d=-2,c=7$
4. If the transformation $5f(2x+6)+9$ of a parent function $f(x)$ is written as $af[k(x-d)]+c$, then:
   
   $a=5,k=2,d=6,c=9$
   $a=5,k=2,d=-6,c=9$
   $a=5,k=2,d=-3,c=9$
   $a=5,k=1,d=-3,c=9$
5. If $f(x)$ is reflected in the $x$-axis and then reflected in the $y$-axis, the resulting function is $\cdots$?
   
   $-f(x)$
   $-f(2x)$
   $-f(\frac{1}{x})$
   $-f(-x)$
6. Let $f(x)=x^2$. Find the image of the point $(2,4)$ under the transformation $-2f(2x)+8$.
   
   $(1,0)$
   $(4,0)$
   $(4,12)$
   $(-4,-8)$
7. Let $f(x)=\sqrt{x}$. Find the image of the point $(9,3)$ after the transformation $f(3x+6)-5$
   
   $(1,-2)$
   $(9,-2)$
   $(33,-2)$
   $(28,10)$
8. Let $f(x)$ be a function. In the transformation $2f[3(x+4)]+5$, the role of $3$ is $\cdots$?
   
   a vertical compression by a factor of $\frac{1}{3}$
   a horizontal compression by a factor of $\frac{1}{3}$
   a horizontal stretch by a factor of $3$
   a horizontal translation of $3$ units to the right
9. Let $f(x)$ be a function. In the transformation $5f\left[\frac{2}{3}(x+4)\right]+9$, the role of $\frac{2}{3}$ is $\cdots$?
   
   a horizontal compression by a factor of $\frac{2}{3}$
   a horizontal stretch by a factor of $2$
   a horizontal compression by a factor of $3$
   a horizontal stretch by a factor of $\frac{3}{2}$
10. Let $f(x)$ be a function. In the transformation $7f\left[-5(x+4)\right]+9$, the $-5$ indicates $\cdots$?
    
    a vertical stretch by a factor of $5$ and a reflection in the $x$-axis
    a horizontal stretch by a factor of $5$ and a reflection in the $y$-axis
    a horizontal compression by a factor of $\frac{1}{5}$ and a reflection in the $y$-axis
    a horizontal compression by a factor of $\frac{1}{5}$ and a reflection in the $x$-axis
11. If $f(x)$ is reflected in both axes and then translated $1$ unit to the right and $5$ units down, the resulting function is $\cdots$?
    
    $-f[-(x+1)]-5$
    $-f[-(x+1)]+5$
    $-f[-(x-5)]-1$
    $-f[-(x-1)]-5$
12. If $f(x)$ is transformed to $af[k(x-d)]+c$, state the mapping rule.
    
    $x\mapsto \frac{x}{k}+d,~y\mapsto ay+c$
    $x\mapsto \frac{x}{k}-d,~y\mapsto ay+c$
    $x\mapsto \frac{k}{x}+d,~y\mapsto ay-c$
    $x\mapsto \frac{x}{k}+d,~y\mapsto \frac{y}{a}+c$
13. Let $f(x)$ be a function such that the image of the point $(2,4)$ under the transformation $af(x)+b$ is $(2,8)$. Find possible $a$ and $b$.
    
    $a=4,b=0$
    $a=2,b=2$
    $a=2,b=0$
    $a=4,b=1$
14. Let $f(x)$ be such that the image of $(a,b)$ under the transformation $3f(x-1)+5$ is $(6,8)$. Find the values of $a$ and $b$.
    
    $a=7,b=3$
    $a=7,b=1$
    $a=1,b=5$
    $a=5,b=1$
15. Find the image of the point $(2,\frac{1}{2})$ under a transformation $-f[2(x-1)]-3$ of $f(x)=\frac{1}{x}$.
    
    $\left(2,-\frac{5}{2}\right)$
    $\left(3,-\frac{7}{2}\right)$
    $\left(2,-\frac{7}{2}\right)$
    $\left(2,\frac{7}{2}\right)$
16. Let $f(x)=x^2$ undergo the transformation $2f(x+3)-4$. The resulting expression is $\cdots$?
    
    $2x^2+12x+18$
    $2x^2+12x+22$
    $2x^2+12x+14$
    $2x^2-12x+14$
17. If $f(x)=\sqrt{x}$ is transformed to $-4f[3x+3]-5$, the domain of the transformed function is $\cdots$?
    
    $\{x\in\mathbb{R}~:~x\geq -3\}$
    $\{x\in\mathbb{R}~:~x\leq -1\}$
    $\{x\in\mathbb{R}~:~x\geq -5\}$
    $\{x\in\mathbb{R}~:~x\geq -1\}$
18. If $f(x)=\sqrt{x}$ is transformed to $-4f[3x+3]-5$, the range of the transformed function is $\cdots$?
    
    $\{y\in\mathbb{R}~:~y\leq -4\}$
    $\{y\in\mathbb{R}~:~y\leq -5\}$
    $\{y\in\mathbb{R}~:~y\geq -5\}$
    $\{y\in\mathbb{R}~:~y\geq -4\}$
19. Let $f(x)=x^2$. Under which of the transformations below is the point $(-1,1)$ invariant?
    
    $-2f\left[\frac{1}{2}(x+1)\right]+3$
    $-2f\left[2(x-1)\right]+3$
    $-2f\left[\frac{1}{2}(x-1)\right]-3$
    $-2f\left[\frac{1}{2}(x-1)\right]+3$
20. Let $f(x)=|x|$. Find the image of the point $(-2,2)$ under the transformation $-f(-x)$.
    
    $(2,0)$
    $(2,2)$
    $(2,-2)$
    $(-2,2)$