Line equation of $ff$

In order to determine the line equation of $ff$, first read off two points from the diagram that lie on the straight line $ff$. In the above example, you may for instance get $A(0\mathrm{\mid }3)A(0|3)$ and $B(4\mathrm{\mid }2)B(4|2)$. Use these two points to determine the slope of $ff$ via the difference quotient

${\displaystyle m_f=\frac{y_B-y_A}{x_B-x_A}}\backslash displaystyle\; m\_f=\backslash frac\{y\_B-y\_A\}\{x\_B-x\_A\}$

Plug in the values

${\displaystyle m_f=\frac{2-3}{4-0}=-\frac{1}{4}}\backslash displaystyle\; m\_f=\backslash frac\{2-3\}\{4-0\}=-\backslash frac14$

Now determine the $yy$-axis intercept $b_{f}b\_f$ by plugging in any point on $ff$ into the general line equation. Alternatively, you may directly read off at which value $ff$ intersects the $yy$-axis (if possible).

$f(x):y=m_{f}x+b_{f}f(x):y=m\_fx+b\_f$

For instance, plug in $AA$.

${\displaystyle 3=-\frac{1}{4}\cdot 0+b_f}\backslash displaystyle\; 3=-\backslash frac14\backslash cdot0+b\_f$

Simplify.

$3=b_f\Rightarrow b_f=33=b\_f\backslash Rightarrow\; b\_f=3$

So the line equation is ${\displaystyle f(x)=-\frac{1}{4}\cdot x+3}\backslash displaystyle\; f(x)=-\backslash frac14\backslash cdot\; x+3$.

Line equation of $gg$

In order to determine the line equation of $gg$, first read off two points from the diagram that lie on the straight line $gg$. In the above example, you may for instance get $C(-4\mathrm{\mid }0)C(-4|0)$ and $D(0\mathrm{\mid }1)D(0|1)$. Use these two points to determine the slope of $gg$ via the difference quotient

${\displaystyle m_g=\frac{y_D-y_C}{x_D-x_C}}\backslash displaystyle\; m\_g=\backslash frac\{y\_D-y\_C\}\{x\_D-x\_C\}$

Plug in the values

${\displaystyle m_g=\frac{1-0}{0-(-4)}=\frac{1}{4}}\backslash displaystyle\; m\_g=\backslash frac\{1-0\}\{0-(-4)\}=\backslash frac14$

Now determine the $yy$-axis intercept $b_{g}b\_g$ by plugging in any point on $gg$ into the general line equation. Alternatively, you may directly read off at which value $gg$ intersects the $yy$-axis (if possible).

$g(x):y=m_{g}x+b_{g}g(x):y=m\_gx+b\_g$

For instance, plug in $DD$.

${\displaystyle 1=\frac{1}{4}\cdot 0+b_g}\backslash displaystyle\; 1=\backslash frac14\backslash cdot0+b\_g$

Simplify.

${\displaystyle 1=b_g\Rightarrow b_g=1}\backslash displaystyle\; 1=b\_g\backslash Rightarrow\; b\_g=1$

So the line equation is ${\displaystyle g(x)=\frac{1}{4}\cdot x+1}\backslash displaystyle\; g(x)=\backslash frac14\backslash cdot\; x+1$.

Line equation of $hh$

In order to determine the line equation of $hh$, first read off two points from the diagram that lie on the straight line $hh$. In the above example, you may for instance get $E(-1\mathrm{\mid }0)E(-1|0)$ and $A(0\mathrm{\mid }3)A(0|3)$. Use these two points to determine the slope of $hh$ via the difference quotient

${\displaystyle m_h=\frac{y_A-y_E}{x_A-x_E}}\backslash displaystyle\; m\_h=\backslash frac\{y\_A-y\_E\}\{x\_A-x\_E\}$

Plug in the values

${\displaystyle m_h=\frac{3-0}{0-(-1)}=3}\backslash displaystyle\; m\_h=\backslash frac\{3-0\}\{0-(-1)\}=3$

Now determine the $yy$-axis intercept $b_{h}b\_h$ by plugging in any point on $hh$ into the general line equation. Alternatively, you may directly read off at which value $hh$ intersects the $yy$-axis (if possible).

$h(x):y=m_{h}x+b_{h}h(x):y=m\_hx+b\_h$

For instance, plug in $AA$.

$3=3\cdot 0+b_{h}3=3\backslash cdot0+b\_h$

Simplify.

$3=b_h\Rightarrow b_h=33=b\_h\backslash Rightarrow\; b\_h=3$

So the line equation is ${\displaystyle h(x)=3\cdot x+3}\backslash displaystyle\; h(x)=3\backslash cdot\; x+3$.

Line equation of $ii$

In order to determine the line equation of $gg$, first read off two points from the diagram that lie on the straight line $gg$. In the above example, you may for instance get $F(0\mathrm{\mid }-3)F(0|-3)$ and $S(6\mathrm{\mid }0)S(6|0)$. Use these two points to determine the slope of $gg$ via the difference quotient

${\displaystyle m_i=\frac{y_S-y_F}{x_S-x_F}}\backslash displaystyle\; m\_i=\backslash frac\{y\_S-y\_F\}\{x\_S-x\_F\}$

Plug in the values

${\displaystyle m_i=\frac{0-(-3)}{6-0}=\frac{1}{2}}\backslash displaystyle\; m\_i=\backslash frac\{0-(-3)\}\{6-0\}=\backslash frac12$

Now determine the $yy$-axis intercept $b_{i}b\_i$ by plugging in any point on $ii$ into the general line equation. Alternatively, you may directly read off at which value $ii$ intersects the $yy$-axis (if possible).

$i(x):y=m_{i}x+b_{i}i(x):y=m\_ix+b\_i$

For instance, plug in $FF$.

${\displaystyle -3=\frac{1}{2}\cdot 0+b_i}\backslash displaystyle-3=\backslash frac12\backslash cdot0+b\_i$

Simplify.

$-3=b_i\Rightarrow b_i=-3-3=b\_i\backslash Rightarrow\; b\_i=-3$

So the line equation is ${\displaystyle i(x)=\frac{1}{2}\cdot x-3}\backslash displaystyle\; i(x)=\backslash frac12\backslash cdot\; x-3$.

Intersection point $P(x_p\mathrm{\mid }{y}_p)P(x\_p|y\_p)$ of $gg$ and $hh$

To determine the point of intersection of two functions, you set them equal and solve for $xx$. The function equations (determined in the previous problem) are $g(x):y={\displaystyle \frac{1}{4}x+1}g(x):y=\backslash displaystyle\backslash frac14x+1$ and $h(x):y=3x+3h(x):y=3x+3$.

Now substitute $-\frac{8}{11}-\backslash frac8\{11\}$ into the line equation of $gg$ or $hh$ to determine $y_{P}y\_P$.

$h(x_P):y_P=3\cdot x_P+3h(x\_P):y\_P=3\backslash cdot\; x\_P+3$

Plug in $x_{P}x\_P$ .

${\displaystyle y_P=3\cdot (-\frac{8}{11})+3=\frac{9}{11}}\backslash displaystyle\; y\_P=3\backslash cdot\backslash left(-\backslash frac8\{11\}\backslash right)+3=\backslash frac9\{11\}$

The lines $gg$ and $hh$ therefore intersect at ${\displaystyle P(-\frac{8}{11}\mid \frac{9}{11})}\backslash displaystyle\; P\backslash left(-\backslash frac8\{11\}\backslash left|\backslash frac9\{11\}\backslash right.\backslash right)$.

You may obtain the zero $x_{N_f}x\_\{N\_f\}$ of $ff$ by setting the function equation ${\displaystyle f(x):y=-\frac{1}{4}x+3}\backslash displaystyle\; f(x):y=-\backslash frac14x+3$ equal to 0 and solving for $xx$.

So $ff$ has the zero 12.

The intersection of $hh$ and $ii$ and the intersection of $gg$ and $ii$ lie outside the figure above.

Intersection point $T(x_T\mathrm{\mid }{y}_T)T(x\_T|y\_T)$ of $hh$ and $ii$

To determine the point of intersection of two functions, you set them equal and solve for $xx$. The function equations (determined in the previous problem) are $h(x):y=3x+3h(x):y=3x+3$ and ${\displaystyle i(x):y=\frac{1}{2}x-3}\backslash displaystyle\; i(x):y=\backslash frac12x-3$.

Now substitute $-\frac{12}{5}-\backslash frac\{12\}5$ into the line equation of $hh$ or $ii$ to determine $y_{T}y\_T$.

$h(x_T):y_T=3\cdot x_T+3h(x\_T):y\_T=3\cdot x\_T+3$

Plug in $x_{T}x\_T$.

${\displaystyle y_T=3\cdot (-\frac{12}{5})+3=-\frac{21}{5}}\backslash displaystyle\; y\_T=3\backslash cdot\backslash left(-\backslash frac\{12\}5\backslash right)+3=-\backslash frac\{21\}5$

The lines $hh$ and $ii$ therefore intersect at ${\displaystyle T(-\frac{5}{2}\mid -\frac{21}{5})}\backslash displaystyle\; T\backslash left(-\backslash frac52\backslash left|-\backslash frac\{21\}5\backslash right.\backslash right)$.

Intersection point $Q(x_Q\mathrm{\mid }{y}_Q)Q(x\_Q|y\_Q)$ of $gg$ and $ii$

To determine the point of intersection of two functions, you set them equal and solve for $xx$. The function equations (determined in the previous problem) are ${\displaystyle g(x):y=\frac{1}{4}x+1}\backslash displaystyle\; g(x):y=\backslash frac14x+1$ and ${\displaystyle i(x):y=\frac{1}{2}x-3}\backslash displaystyle\; i(x):y=\backslash frac12x-3$.

Now substitute $1616$ into the line equation of $gg$ or $ii$ to determine $y_{Q}y\_Q$.

${\displaystyle g(x_Q):y_Q=\frac{1}{4}\cdot x_Q+1}\backslash displaystyle\; g(x\_Q):y\_Q=\backslash frac14\cdot x\_Q+1$

Plug in $x_{Q}x\_Q$.

${\displaystyle y_Q=\frac{1}{4}\cdot 16+1=5}\backslash displaystyle\; y\_Q=\backslash frac14\backslash cdot16+1=5$

The lines $gg$ and $ii$ therefore intersect at ${\displaystyle Q(16\mathrm{\mid }5)}\backslash displaystyle\; Q(16|5)$.

If no pair of lines is parallel, then there are a total of 6 intersections, namely the following:

• $ff$ and $gg$

• $ff$ and $hh$

• $ff$ and $ii$

• $gg$ and $hh$

• $gg$ and $ii$

• $hh$ and $ii$