Exercises: Linear functions and line equations

Determining function equations.

A line has the gradient $a_{1}$ and passes through the point $P$ . Determine the equation of the function $f (x)$ , the points of intersection and draw the graph.

$a_{1} = \frac{1}{2}$ $P (4 | - 2)$

For this task you need the following basic knowledge: Linear function
Determining the function equaition
General line equation: $y = m \cdot x + t$
here $m = a_{1}$
$f (x)$ $=$ $a_{1} \cdot x + t$
↓
plug $a_{1} = \frac{1}{2}$ into the general line equation.
$f (x)$ $=$ $\frac{1}{2} x + t$
↓
plug $P$ into $f (x)$
$- 2$ $=$ $\frac{1}{2} \cdot 4 + t$ $- 2$
↓
solve for $t$
$t$ $=$ $- 4$
↓
Plug $t$ into $f (x)$ .
$=$
$f (x) = \frac{1}{2} x - 4$
Determining the $y$ -axis intercept
What we are looking for is the so-called y-axis intercept (here: $t$ ), i.e. where $y = f (0) = 0$ and $x = 0$ .
Since the general equation of a straight line is
$f (x) = m \cdot x + t$ , we have
$f (0) = m \cdot 0 + t = t$ .
Here $t = - 4$
$\Rightarrow$ Intersection with the $y$ -axis at $(0 | - 4)$
Determining the $x$ -axis intercept
$f (x)$ $=$ $0$
↓
We are looking for an $x$ with $f (x) = 0$ .
$\frac{1}{2} x - 4$ $=$ $0$ $+ 4$
$\frac{1}{2} x$ $=$ $4$ $: \frac{1}{2}$
$x$ $=$ $\frac{4}{\frac{1}{2}}$
↓
You divide by a fraction $\to$ multiply by the reciprocal.
$x$ $=$ $4 \cdot 2$
$x$ $=$ $8$
$\Rightarrow$ Intersection with the $x$ -axis at $(8 | 0)$
Plot

$a_{1} = \frac{3}{4} P (1 | - 3)$

$a_{1} = 2 P (3 | - 1)$

$a_{1} = \frac{4}{5} P (\frac{3}{2} | 4)$

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Exercises: Linear functions and line equations

Determining the function equaition

Determining the $y$ -axis intercept

Determining the $x$ -axis intercept

Plot

Determining the function equation

Determining the axis intercepts

Plot

Determining the function equation

Determining the axis intercepts

Plot

Determining the function equation

Determining the axis intercepts

Plot

$f (x)$	$=$	$a_{1} \cdot x + t$
	↓	plug $a_{1} = \frac{1}{2}$ into the general line equation.
$f (x)$	$=$	$\frac{1}{2} x + t$
	↓	plug $P$ into $f (x)$
$- 2$	$=$	$\frac{1}{2} \cdot 4 + t$	$- 2$
	↓	solve for $t$
$t$	$=$	$- 4$
	↓	Plug $t$ into $f (x)$ .
	$=$

$f (x)$	$=$	$0$
	↓	We are looking for an $x$ with $f (x) = 0$ .
$\frac{1}{2} x - 4$	$=$	$0$	$+ 4$
$\frac{1}{2} x$	$=$	$4$	$: \frac{1}{2}$
$x$	$=$	$\frac{4}{\frac{1}{2}}$
	↓	You divide by a fraction $\to$ multiply by the reciprocal.
$x$	$=$	$4 \cdot 2$
$x$	$=$	$8$

$f (x)$	$=$	$a_{1} \cdot x + t$
	↓	$t$ : $y$ -axis intercept Plug $a_{2} = \frac{3}{4}$ into the general line equation.
$f (x)$	$=$	$\frac{3}{4} x + t$
	↓	Plug $P (1 \| - 3)$ into $f (x)$ .
$- 3$	$=$	$\frac{3}{4} \cdot 1 + t$
	↓	multiply
$- 3$	$=$	$\frac{3}{4} + t$	$- \frac{3}{4}$
$t$	$=$	$- 3 - \frac{3}{4}$
	↓	add
$t$	$=$	$- 3.75$
	↓	Plug $t$ into $f (x)$ .
	$=$

$\frac{3}{4} x - 3.75$	$=$	$0$	$+ 3.75$
$\frac{3}{4} x$	$=$	$3.75$	$: \frac{3}{4}$
$x$	$=$	$3.75 : \frac{3}{4}$
	↓	You divide by a fraction $\to$ multiply by the reciprocal.
$x$	$=$	$3.75 \cdot \frac{4}{3}$
	↓	multiply
$x$	$=$	$5$

$f (x)$	$=$	$a_{1} \cdot x + t$
	↓	$t$ : $y$ -axis intercept Plug $a_{3} = 2$ into the general line equation.
$f (x)$	$=$	$2 x + t$
	↓	Plug $P (3 \| - 1)$ into $f (x)$ .
$- 1$	$=$	$2 \cdot 3 + t$
	↓	multiply
$- 1$	$=$	$6 + t$	$- 6$
$t$	$=$	$- 1 - 6$
	↓	subtract
$t$	$=$	$- 7$
	↓	Plug $t$ into $f (x)$ .
	$=$

$2 x - 7$	$=$	$0$	$+ 7$
$2 x$	$=$	$7$	$: 2$
$x$	$=$	$7 : 2$
	↓	divide
$x$	$=$	$3.5$

$f (x)$	$=$	$a_{1} \cdot x + t$
	↓	$t$ : $y$ -axis intercept Plug $a_{4} = \frac{4}{5}$ into the general line equation
$f (x)$	$=$	$\frac{4}{5} x + t$
	↓	Plug $P (\frac{3}{2} \| 4)$ into $f (x)$ .
$4$	$=$	$\frac{4}{5} \cdot \frac{3}{2} + t$
	↓	shorten by 2
$4$	$=$	$\frac{2}{5} \cdot 3 + t$
	↓	multiply
$4$	$=$	$\frac{6}{5} + t$	$- \frac{6}{5}$
$t$	$=$	$4 - \frac{6}{5}$
	↓	Write 4 as a fraction with a 4 in the denominator
$t$	$=$	$\frac{20}{5} - \frac{6}{5}$
	↓	subtract
$t$	$=$	$\frac{14}{5}$
$t$	$=$	$2.8$
	↓	Setze t in f(x) ein.

$\frac{4}{5} x + 2.8$	$=$	$0$	$- 2.8$
$\frac{4}{5} x$	$=$	$- 2.8$	$: \frac{4}{5}$
$x$	$=$	$- 2.8 : \frac{4}{5}$
	↓	divide
$x$	$=$	$- \frac{7}{2}$
$x$	$=$	$- 3.5$

Exercises: Linear functions and line equations

Determining the function equaition

Determining the y-axis intercept

Determining the x-axis intercept

Plot

Determining the function equation

Determining the axis intercepts

Plot

Determining the function equation

Determining the axis intercepts

Plot

Determining the function equation

Determining the axis intercepts

Plot

Determining the $y$ -axis intercept

Determining the $x$ -axis intercept