Linear function

Straight line graph

1. ?
   ?

   Equation of a linear function is usually written in the slope-intercept form (called also explicit form):
   
   $f(x)=mx+c$     or     $y=mx+c$
   Write down the gradient $(m)$ and $y$-intercept $(c)$ for the following linear functions:
   

   (a)   $f(x)=2x-5$

   (b)   $f(x)=x+3$

   (c)   $f(x)=3-x$

   (d)   $f(x)=5$

   Solutions:    (a)  $m=2,c=-5$;     (b)  $m=1,c=3$;     (c)  $m=-1,c=3$;     (d)  $m=0,c=5$
2. Write down the gradient $(m)$ and $y$-intercept $(c)$ for the following straight lines:
   

   (a)   $y=-x$

   (b)   $y=\frac{1}{2}x+3$

   (c)   $y=\frac{1}{2}+\frac{x}{3}$

   (d)   $y=\frac{3x-2}{12}$

   Solutions:    (a)  $m=-1,c=0$;     (b)  $m=\frac{1}{2},c=3$;     (c)  $m=\frac{1}{3},c=\frac{1}{2}$;     (d)  $m=\frac{1}{4},c=-\frac{1}{6}$
3. Write down the gradient $(m)$ and $y$-intercept $(c)$ for the following straight lines:
   

   (a)   $x+y=2$

   (b)   $2x-4y+1=0$

   (c)   $\frac{x}{3}+\frac{y}{6}=1$

   Solutions:    (a)  $m=-1,c=2$;     (b)  $m=\frac{1}{2},c=\frac{1}{4}$;     (c)  $m=-2,c=6$
4. Draw the graphs of the following linear functions:
   

   (a)   $f(x)=x+2$

   (b)   $f(x)=2x-1$

   (c)   $f(x)=\frac{1}{3}x+1$

5. ?
   ?

   Equation of a linear function can be written in other forms, too:
   
   Implicit form: $ax+by+c=0$
   
   Double intercept form: $\frac{x\phantom{y}}{a\phantom{b}}+\frac{y}{b}=1$
   Draw the following straight lines:
   

   (a)   $y=x$

   (b)   $y=-2x+3$

   (c)   $x-2y-4=0$

   (d)   $\frac{x}{3}+\frac{y}{2}=1$

6. Straight line has the equation $y=3x-5$. Which of the following points lies on this straight line:
   

   (a)   $A(2,1)$

   (b)   $B(-3,-15)$

   (c)   $C(25,80)$

   (d)   $D(\frac{7}{6},-\frac{3}{2})$

   Solutions:    (a)  lies;     (b)  doesn't lie;     (c)  doesn't lie;     (d)  lies on the given straight line
7. A point has the coordinates $P(3,-2)$. Which of the following lines passes through this point:
   

   (a)   $y=1-x$

   (b)   $2x+3y=0$

   (c)   $2x-5y-13=0$

   (d)   $y=\frac{1}{2}x-\frac{4}{3}$

   Solutions:    (a)  passes;     (b)  passes;     (c)  doesn't pass;     (d)  doesn't pass through the given point
8. ?
   ?

   Equation of a linear function passing through given points $A(x_1,y_1)$ and $B(x_2,y_2)$ can be written using the following formulas:
   
   $m=\frac{y_2-y_1}{x_2-x_1}$    or    $m=\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}$
   
   $y-y_1=m(x-x_1)$
   Straight line $\ell$ passes through points $A(1,-2)$ and $B(4,7)$. Write down the equation of this straight line.
   Solution:    $y=3x-5$
9. Straight line $\ell$ passes through points $A(-2,-1)$ and $B(7,5)$.
   

   (a)   Write down the equation of this straight line.

   (b)   Draw this straight line in the coordinate system.

   Solutions:    (a)  $y=\frac{2}{3}x+\frac{1}{3}$
10. Straight line $\ell$ passes through points $A(\frac{1}{3},-\frac{1}{4})$ and $B(\frac{4}{3},\frac{1}{2})$.
    

    (a)   Write down the equation of this straight line.

    (b)   Find the coordinates of the $x$-axis intercept.

    (c)   Draw this straight line in the coordinate system.

    Solutions:    (a)  $y=\frac{3}{4}x-\frac{1}{2}$;     (b)  $C(\frac{2}{3},0)$
11. Straight line has the equation $4x+3y=24$.
    

    (a)   Find the coordinates of $x$-axis intercept $A$ and $y$-axis intercept $B$.

    (b)   Draw this straight line in the coordinate system.

    (c)   Calculate the length of the line segment $AB$.

    Solutions:    (a)  $A(6,0),B(0,8)$;     (c)  $AB=10$
12. Straight line has the equation $\frac{x}{2}-\frac{y}{1.5}=1$.
    

    (a)   Find the coordinates of $x$-axis intercept $A$ and $y$-axis intercept $B$.

    This straight line and both coordinate axes form a triangle $ABO$.

    (b)   Calculate the perimeter of the triangle $ABO$.

    Solutions:    (a)  $A(2,0),B(0,-\frac{3}{2})$;     (b)  $P=6$
13. Straight line has the equation $y=\frac{x-12}{3}$.
    

    (a)   Find the coordinates of $x$- and $y$-axis intercepts.

    (b)   Calculate the area of the triangle formed by this line and both coordinate axes.

    Solutions:    (a)  $A(12,0),B(0,-4)$;     (b)  $A=24$

Parallel and perpendicular lines

14. ?
    ?

    Condition of parallelism
    Straight lines are parallel if they have equal gradients:
    
    $m_2=m_1$
    Straight line ${\ell }_1$ has the equation $y=3x-4$. Write the equation of the straight line ${\ell }_2$ which is parallel to ${\ell }_1$ and passes through $P(2,7)$.
    Solution:    $y=3x+1$
15. Straight line ${\ell }_1$ has the equation $x-6y+3=0$.
    

    (a)   Find the gradient and the $y$-intercept of the line ${\ell }_1$.

    (b)   Write the equation of the straight line ${\ell }_2$ which is parallel to ${\ell }_1$ and passes through the origin.

    Solutions:    (a)  $m=\frac{1}{6},c=\frac{1}{2}$;     (b)  $y=\frac{1}{6}x$
16. Straight line ${\ell }_1$ has the equation $\frac{x}{6}+\frac{y}{4}=1$.
    

    (a)   Find the gradient of the line ${\ell }_1$.

    (b)   Write the equation of the straight line ${\ell }_2$ which is parallel to ${\ell }_1$ and passes through $P(3,1)$.

    Solutions:    (a)  $m=-\frac{2}{3}$;     (b)  $y=-\frac{2}{3}x+3$
17. Straight lines ${\ell }_1$ and ${\ell }_2$ have the equations    ${\ell }_1:y=\frac{1-2x}{6}$    and    ${\ell }_2:y=mx-2$.
    

    (a)   Find $m$, given that ${\ell }_1||{\ell }_2$.

    (b)   Find the area of the triangle formed by the line ${\ell }_2$ and both coordinate axes.

    Solutions:    (a)  $m=-\frac{1}{3}$;     (b)  $A=6$
18. ?
    ?

    Condition of perpendicularity
    Straight lines are perpendicular if:
    
    $m_2=-\frac{1}{m_1}$
    Straight line ${\ell }_1$ has the equation $y=3x-4$. Write the equation of the straight line ${\ell }_2$ which is perpendicular to ${\ell }_1$ and passes through $P(3,1)$.
    Solution:    $y=-\frac{1}{3}x+2$
19. Straight line ${\ell }_1$ has the equation $8x+6y=5$.
    

    (a)   Find the gradient of the line ${\ell }_1$.

    (b)   Write the equation of the straight line ${\ell }_2$ which is perpendicular to ${\ell }_1$ and passes through $P(2,4)$.

    Solutions:    (a)  $m=-\frac{4}{3}$;     (b)  $y=\frac{3}{4}x+\frac{5}{2}$
20. Straight line ${\ell }_1$ has the equation $y=\frac{4}{3}x-\frac{16}{3}$.
    

    (a)   Write down the coordinates of point $M$ where line ${\ell }_1$ intercepts the $x$-axis.

    Straight line ${\ell }_2$ is perpendicular to ${\ell }_1$ and passes through the same point $M$.

    (b)   Write the equation of the straight line ${\ell }_2$.

    Straight line ${\ell }_2$ together with both coordinate axes forms a triangle $MNO$.

    (c)   Write down the coordinates of point $N$.

    (d)   Calculate the area of the triangle $MNO$.

    (e)   Calculate the perimeter of the triangle $MNO$.

    Solutions:    (a)  $M(4,0)$;     (b)  $y=-\frac{3}{4}x+3$;     (c)  $N(0,3)$;     (d)  $A=6$;     (e)  $P=12$
21. Straight line segment has the endpoints $A(2,2)$ and $B(6,4)$.
    

    (a)   Find the midpoint $M$ of this line segment.

    (b)   Write the equation of the straight line which is perpendicular to this line segment and passes through $M$.

    Solutions:    (a)  $M(4,3)$;     (b)  $y=-2x+11$
22. Straight line segment has the endpoints $A(2,6)$ and $B(4,0)$. Find the equation of the perpendicular bisector of this line segment.
    Solution:    $y=\frac{1}{3}x+2$
23. Find the equation of the perpendicular bisector of the line segment with endpoints $P(1,-2)$ and $R(-3,4)$.
    Solution:    $y=\frac{2}{3}x+\frac{5}{3}$

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