Functions

Transformations of graphs

1. ?
   ?

   Adding a number $q$ to the equation of a function causes the translation in $y$ axis (or vertical shift) of the graph:
   
   $y=f(x)+q$
   Using your GDC draw the graph of the function $y=x^2$. Then use your GDC to draw the following graphs and explain the transformation used in each case:
   

   (a)   $y=x^2+1$

   (b)   $y=x^2+3$

   (c)   $y=x^2-2$

   Solutions:    (a)  Translation of 1 unit parallel to the $y$-axis (shift 1 unit upwards);     (b)  translation of 3 units parallel to the $y$-axis (shift 3 units upwards);     (c)  translation of  −2 units parallel to the $y$-axis (shift 2 units downwards)
2. Using your GDC draw the graph of the function $y=x^3-x$.
   

   (a)   Shift this function 4 units upwards. Write down the equation of the new function and draw the graph.

   (b)   Shift this function 1 unit downwards. Write down the equation of the new function and draw the graph.

   Solutions:    (a)  $y=x^3-x+4$;     (b)  $y=x^3-x-1$
3. ?
   ?

   Subtracting a number $p$ from the unknown $x$ inside the equation of a function causes the translation in $x$ axis (or horizontal shift) of the graph:
   
   $y=f(x-p)$
   Using your GDC draw the graph of the function $y=x^2$. Then use your GDC to draw the following graphs and explain the transformation used in each case:
   

   (a)   $y=(x-1{)}^2$

   (b)   $y=(x-2{)}^2$

   (c)   $y=(x+3{)}^2$

   Solutions:    (a)  Translation of 1 unit parallel to the $x$-axis (shift 1 unit to the right);     (b)  translation of 2 units parallel to the $x$-axis (shift 2 units to the right);     (c)  translation of  −3 units parallel to the $x$-axis (shift 3 units to the left)
4. Using your GDC draw the graph of the function $y=x^3-x$.
   

   (a)   Shift this function 2 units to the right. Write down the equation of the new function and draw the graph.

   (b)   Shift this function 1 unit to the left. Write down the equation of the new function and draw the graph.

   Solutions:    (a)  $y=(x-2{)}^3-(x-2)$;     (b)  $y=(x+1{)}^3-(x+1)$
5. Using your GDC draw the graph of the function $f(x)=x^3-3x+2$. Then use your GDC to draw the following graphs. In each case write down the equation of the new function and explain the transformation used.
   

   (a)   $y=f(x)+1$

   (b)   $y=f(x)-2$

   (c)   $y=f(x-3)$

   (d)   $y=f(x+2)$

   (e)   $y=f(x-2)-3$

   Solutions:    (a)  $y=x^3-3x+3$, shift 1 unit upwards;     (b)  $y=x^3-3x$, shift 2 units downwards;     (c)  $y=(x-3{)}^3-3(x-3)+2$, shift 3 units to the right;     (d)  $y=(x+2{)}^3-3(x+2)+2$, shift 2 units to the left;     (e)  $y=(x-2{)}^3-3(x-2)-1$, shift 2 units to the right and 3 units downwards
6. Let $f(x)=-x^2+4x-3$.
   

   (a)   Using your GDC draw the graph of the function $f$.

   Consider the function $g(x)=f(x+4)+2$

   (b)   Write the equation of the function $g$. Simplify this equation.

   (c)   Use your GDC to draw the graph of the function $g$.

   Solutions:    (b)  $g(x)=-(x+4{)}^2+4(x+4)-3+2$, simplified: $g(x)=-x^2-4x-1$
7. ?
   ?

   Multiplying the equation of a function by a number $a$ causes the dilation in $y$ axis (or vertical stretch) of the graph:
   
   $y=af(x)$
   Using your GDC draw the graph of the function $f(x)=\frac{1}{x^2+1}$. Then use your GDC to draw the following graphs and explain the transformation used in each case:
   

   (a)   $y=2f(x)$

   (b)   $y=3f(x)$

   (c)   $y=\frac{1}{2}f(x)$

   Solutions:    (a)  Dilation along the $y$-axis by a scale factor 2 (stretch in $y$ by 2);     (b)  dilation along the $y$-axis by a scale factor 3 (stretch in $y$ by 3);     (c)  dilation along the $y$-axis by a scale factor $\frac{1}{2}$ (stretch in $y$ by $\frac{1}{2}$)
8. Using your GDC draw the graph of the function $f(x)=\frac{1}{x^2+1}$. Then use your GDC to draw the following graphs and explain the transformation used in each case:
   

   (a)   $y=-f(x)$

   (b)   $y=-2f(x)$

   (c)   $y=-\frac{1}{2}f(x)$

   Solutions:    (a)  Dilation along the $y$-axis by a scale factor  −1  =  reflection in the $x$-axis (flip upside down);     (b)  dilation along the $y$-axis by a scale factor  −2 (flip upside down and stretch in $y$ by 2);     (c)  dilation along the $y$-axis by a scale factor $-\frac{1}{2}$ (flip upside down and stretch in $y$ by $\frac{1}{2}$)
9. Let $f(x)=x^2-4x$.
   

   (a)   Using your GDC draw the graph of the function $f$.

   Consider the function $g(x)=\frac{1}{2}f(x)$

   (b)   Write the equation of the function $g$. Simplify this equation.

   (c)   Use your GDC to draw the graph of the function $g$.

   Solutions:    (b)  $g(x)=\frac{1}{2}(x^2-4x)$, simplified: $g(x)=\frac{1}{2}{x}^2-2x$
10. ?
    ?

    Dividing the unknown $x$ inside the equation of a function by a number $b$ causes the dilation in $x$ axis (or horizontal stretch) of the graph:
    
    $y=f(\frac{x}{b})$
    Using your GDC draw the graph of the function $f(x)=x^4$. Then use your GDC to draw the following graphs and explain the transformation used in each case:
    

    (a)   $y=f(\frac{x}{2})$

    (b)   $y=f(\frac{x}{3})$

    (c)   $y=f(2x)$

    Solutions:    (a)  Dilation along the $x$-axis by a scale factor 2 (stretch in $x$ by 2);     (b)  dilation along the $x$-axis by a scale factor 3 (stretch in $x$ by 3);     (c)  dilation along the $x$-axis by a scale factor $\frac{1}{2}$ (stretch in $x$ by $\frac{1}{2}$)
11. Using your GDC draw the graph of the function $f(x)=2^x$. Then use your GDC to draw the following graphs and explain the transformation used in each case:
    

    (a)   $y=f(-x)$

    (b)   $y=f(-2x)$

    (c)   $y=f(-\frac{x}{2})$

    Solutions:    (a)  Dilation along the $x$-axis by a scale factor  −1  =  reflection in the $y$-axis (flip left-to-right);     (b)  dilation along the $x$-axis by a scale factor $-\frac{1}{2}$ (flip left-to-right and stretch in $x$ by $\frac{1}{2}$);     (c)  dilation along the $x$-axis by a scale factor  −2 (flip left-to-right and stretch in $x$ by 2)
12. The graph of the function $y=f(x)$ is drawn in the coordinate system below. Draw the following graphs using transformations:
    

    (a)   $y=f(x)-2$

    (b)   $y=f(x-2)$

    (c)   $y=2f(x)$

    (d)   $y=-\frac{1}{2}f(x)$

    (e)   $y=f\left(\frac{1}{2}x\right)$

    

    Solutions:    (a)  Translation of  −2 units along $y$-axis;     (b)  translation of 2 units along $x$-axis;     (c)  dilation along the $y$-axis by a scale factor 2;     (d)  dilation along the $y$-axis by a scale factor $-\frac{1}{2}$;     (e)  dilation along the $x$-axis by a scale factor 2
13. Function $f$ has the equation $f(x)=\frac{2x}{x^2+1}$. Graph of this function is drawn in the coordinate system below (green graph). In the same coordinate system you can find the graph of the function $g$ (red graph).
    

    

    (a)   Describe transformations required to transform $f$ to $g$.

    (b)   Write the equation of the function $g$.

    (c)   Use your GDC to verify the equation you've just written.

    Solutions:    (a)  Flip upside-down followed by a vertical shift by 3 units;     (b)  $g(x)=3-\frac{2x}{x^2+1}$

Inverse function

14. ?
    ?

    Inverse function is a function which operates in the opposite way as the given function:
    
    $f^{-1}(a)=b$    if    $f(b)=a$
    
    To find the equation of the inverse function you must start with the equation of the original function and swap $x$ and $y$.
    Find the inverse of each of the following functions:
    

    (a)   $f(x)=2x+1$

    (b)   $f(x)=3x-2$

    (c)   $f(x)=\frac{1}{4}x+\frac{1}{2}$

    Solutions:    (a)  $f^{-1}(x)=\frac{x-1}{2}$;     (b)  $f^{-1}(x)=\frac{x+2}{3}$;     (c)  $f^{-1}(x)=4x-2$
15. Find the inverse of each of the following functions:
    

    (a)   $f(x)=x^3+2$

    (b)   $f(x)=2x^3$

    (c)   $f(x)=\frac{1}{4}{x}^3$

    Solutions:    (a)  $f^{-1}(x)=\sqrt[3]{x-2}$;     (b)  $f^{-1}(x)=\sqrt[3]{\frac{x}{2}}$;     (c)  $f^{-1}(x)=\sqrt[3]{4x}$
16. Find the inverse of each of the following functions:
    

    (a)   $f(x)=2^x$

    (b)   $f(x)=2^x+1$

    (c)   $f(x)=2^{x+1}$

    Solutions:    (a)  $f^{-1}(x)={\log }_{2}x$;     (b)  $f^{-1}(x)={\log }_2(x-1)$;     (c)  $f^{-1}(x)={\log }_{2}x-1$
17. Given the function $f(x)=\sqrt[3]{x+3}$
    

    (a)   write down the inverse function $f^{-1}(x)$

    (b)   draw graphs of $f(x)$ and $f^{-1}(x)$ in the same coordinate system.

    Solutions:    (a)  $f^{-1}(x)=x^3-3$
18. Find the inverse of each of the following functions:
    

    (a)   ${\displaystyle f(x)=\frac{x+5}{3x-2}}$

    (b)   ${\displaystyle f(x)=\frac{3x}{x-2}}$

    Solutions:    (a)  $f^{-1}(x)=\frac{2x+5}{3x-1}$;     (b)  $f^{-1}(x)=\frac{2x}{x-3}$
19. Find the inverse of each of the following functions:
    

    (a)   ${\displaystyle f(x)=\frac{x-5}{x+2}}$

    (b)   ${\displaystyle f(x)=\frac{1+x}{1-x}}$

    Solutions:    (a)  $f^{-1}(x)=\frac{2x+5}{-x+1}$;     (b)  $f^{-1}(x)=\frac{x-1}{x+1}$
20. Find the inverse of each of the following functions:
    

    (a)   ${\displaystyle f(x)=\frac{x+2}{x-1}}$

    (b)   ${\displaystyle f(x)=\frac{2x}{x-2}}$

    Solutions:    (a)  $f^{-1}(x)=\frac{x+2}{x-1}$;     (b)  $f^{-1}(x)=\frac{2x}{x-2}$    —  in both cases $f^{-1}(x)=f(x)$
21. Prove that the function ${\displaystyle f(x)=\frac{3x+1}{x-3}}$ is its own inverse.
    Solutions:    Show that $f^{-1}(x)=f(x)$
22. Given the function ${\displaystyle f(x)=\frac{2x-5}{3x-4}}$ write down the following functions:
    

    (a)   $f(x^{-1})$

    (b)   $(f(x){)}^{-1}$

    (c)   $f^{-1}(x)$

    Solutions:    (a)  $f(x^{-1})=\frac{5x-2}{4x-3}$;     (b)  $(f(x){)}^{-1}=\frac{3x-4}{2x-5}$;     (c)  $f^{-1}(x)=\frac{4x-5}{3x-2}$
23. Find the inverse of the function $f(x)=x^2-3$
    

    (a)   for $x⩾0$,

    (b)   for $x⩽0$.

    Solutions:    (a)  $f^{-1}(x)=\sqrt{x+3}$;     (b)  $f^{-1}(x)=-\sqrt{x+3}$
24. Find the inverse of the function $f(x)=\sqrt{x^2+4}$ for $x⩾0$. Write down the domain and range of $f^{-1}$.
    Solutions:    $f^{-1}(x)=\sqrt{x^2-4}$,   domain $=[2,\mathrm{\infty })$,   range $=[0,\mathrm{\infty })$
25. Find the inverse of the function $f(x)=\frac{4}{x^2+1}$ for $x⩾0$. Write down the domain and range of $f^{-1}$.
    Solutions:    $f^{-1}(x)=\sqrt{\frac{4}{x}-1}$,   domain $=(0,4]$,   range $=[0,\mathrm{\infty })$

 Index