Linear function

Straight line graph

1. ?
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   Equation of a linear function is usually written in the slope-intercept form (called also explicit form):
   
   \(f(x)=m\,x+c\)     or     \(y=m\,x+c\)
   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)   \(y=\frac{\textstyle 1}{\textstyle 2}+\frac{\textstyle x}{\textstyle 3}\)

   (d)   \(y=\frac{\textstyle 3x-2}{\textstyle 12}\)

   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)   \(\frac{\textstyle x}{\textstyle 3}+\frac{\textstyle y}{\textstyle 6}=1\)

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

   (a)   \(y=x\)

   (b)   \(y=-2x+3\)

   (c)   \(x-2y-4=0\)

   (d)   \(\frac{\textstyle x}{\textstyle 3}+\frac{\textstyle y}{\textstyle 2}=1\)

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:
   
   \(m=\frac{\textstyle y_2-y_1}{\textstyle x_2-x_1}\)    or    \(m=\frac{\textstyle \Delta y}{\textstyle \Delta x}\)
   
   \(y-y_1=m\, (x-x_1)\)
   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 \(\frac{\textstyle x}{\textstyle 2}-\frac{\textstyle y}{\textstyle 1.5}=1\).
    

    (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 \(y=\frac{\textstyle x-12}{\textstyle 3}\).
    

    (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. ?
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    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 \(\frac{\textstyle x}{\textstyle 6}+\frac{\textstyle y}{\textstyle 4}=1\).
    

    (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    \(\ell_1\!:~~ y=\frac{\textstyle1-2x}{\textstyle 6}\)    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:
    
    \(m_2=-\,\frac{\textstyle 1}{\textstyle m_1}\)
    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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