Exercises: Linear functions and line equations

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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}\left(4|-2\right)\$

Für diese Aufgabe benötigst Du folgendes Grundwissen: Linear function

Determining the function equaition

General line equation: $y=m\cdot x+t$

here $m=a_{1}$

$\displaystyle f\left(x\right)$	$=$	$\displaystyle {a}_1\cdot{x}+{t}$
	↓	plug $a_1 =\tfrac{1}{2}$ into the general line equation.
$\displaystyle f\left(x\right)$	$=$	$\displaystyle \frac{1}{2}x+t$
	↓	plug $P$ into $f (x)$
$\displaystyle -2$	$=$	$\displaystyle \frac{1}{2}\cdot4+t$	$\displaystyle -2$
	↓	solve for $t$
$\displaystyle t$	$=$	$\displaystyle -4$
	↓	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

$f(0)=m\cdot0+t=t$ .

Here $t = - 4$

$\Rightarrow$ Intersection with the $y$ -axis at $\left(0|-4\right)$

Determining the $x$ -axis intercept

$\displaystyle f\left(x\right)$	$=$	$\displaystyle 0$
	↓	We are looking for an $x$ with $f (x) = 0$ .
$\displaystyle \frac{1}{2}x-4$	$=$	$\displaystyle 0$	$\displaystyle +4$
$\displaystyle \frac{1}{2}x$	$=$	$\displaystyle 4$	$\displaystyle :\frac{1}{2}$
$\displaystyle x$	$=$	$\displaystyle \frac{4}{\frac{1}{2}}$
	↓	You divide by a fraction $\rightarrow$ multiply by the reciprocal.
$\displaystyle x$	$=$	$\displaystyle 4\cdot2$
$\displaystyle x$	$=$	$\displaystyle 8$

$\Rightarrow$ Intersection with the $x$ -axis at $\left(8|0\right)$

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${a}_1=\frac34\;\;\;\;\;\;\;\;\;\;\;P\left( 1| -3\right)$

Für diese Aufgabe benötigst Du folgendes Grundwissen: Linear function

Determining the function equation

General line equation:

$\displaystyle f\left(x\right)$	$=$	$\displaystyle a_1\cdot x+t$
	↓	$t$ : $y$ -axis intercept Plug $a_2=\frac34$ into the general line equation.
$\displaystyle f\left(x\right)$	$=$	$\displaystyle \frac{3}{4}\mathrm{x}+\mathrm{t}$
	↓	Plug $P (1 ∣ - 3)$ into $f (x)$ .
$\displaystyle -3$	$=$	$\displaystyle \frac{3}{4}\cdot1+t$
	↓	multiply
$\displaystyle -3$	$=$	$\displaystyle \frac{3}{4}+t$	$\displaystyle -\frac{3}{4}$
$\displaystyle t$	$=$	$\displaystyle -3-\frac{3}{4}$
	↓	add
$\displaystyle t$	$=$	$\displaystyle -3.75$
	↓	Plug $t$ into $f (x)$ .
	$=$

$f(x)=\frac{3}{4}x-3.75$

Determining the axis intercepts

Set $f (x) = 0$ , to obtain the zeros of the function.

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

Intersection with the $x$ -axis at $(5 ∣ 0)$

The $y$ -axis intercept corresponds to the point of intersection with the $y$ -axis.

Intersection with the $y$ -axis at $(0 ∣ - 3.75)$

Plot

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${a}_1=2\;\;\;\;\;\;\;\;\;\;P\left(3|-1\right)$

Für diese Aufgabe benötigst Du folgendes Grundwissen: Linear function

Determining the function equation

General line equation: Here $m = a_{2}$

$\displaystyle f\left(x\right)$	$=$	$\displaystyle a_1\cdot x+t$
	↓	$t$ : $y$ -axis intercept Plug $a_{3} = 2$ into the general line equation.
$\displaystyle f\left(x\right)$	$=$	$\displaystyle 2x+t$
	↓	Plug $P (3 ∣ - 1)$ into $f (x)$ .
$\displaystyle -1$	$=$	$\displaystyle 2\cdot3+t$
	↓	multiply
$\displaystyle -1$	$=$	$\displaystyle 6+t$	$\displaystyle -6$
$\displaystyle t$	$=$	$\displaystyle -1-6$
	↓	subtract
$\displaystyle t$	$=$	$\displaystyle -7$
	↓	Plug $t$ into $f (x)$ .
	$=$

${f}({x})=2\mathrm{x}-7$

Determining the axis intercepts

Set $f (x) = 0$ , to obtain the zeros of the function.

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

$\Rightarrow$ Intersection with the $x$ -axis at $(\frac 7 2|0)$ .

The $y$ -axis intercept corresponds to the point of intersection with the $y$ -axis.

$\Rightarrow$ Intersection with the $y$ -axis at $(0 ∣ - 7)$

Plot

Geogebra File: https://assets.serlo.org/legacy/6350_ufOpT5liEV.xml

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${a}_1=\frac45\;\;\;\;\;\;\;\;\;\;P\left(\frac32|4\right)$

Für diese Aufgabe benötigst Du folgendes Grundwissen: Linear function

Determining the function equation

General line equation:

$\displaystyle f\left(x\right)$	$=$	$\displaystyle a_1\cdot x+t$
	↓	$t$ : $y$ -axis intercept Plug $a_4=\frac45$ into the general line equation
$\displaystyle f\left(x\right)$	$=$	$\displaystyle \frac{4}{5}x+t$
	↓	Plug $P\left(\frac32\|4\right)$ into $f (x)$ .
$\displaystyle 4$	$=$	$\displaystyle \frac{4}{5}\cdot\frac{3}{2}+t$
	↓	shorten by 2
$\displaystyle 4$	$=$	$\displaystyle \frac{2}{5}\cdot3+t$
	↓	multiply
$\displaystyle 4$	$=$	$\displaystyle \frac{6}{5}+t$	$\displaystyle -\frac{6}{5}$
$\displaystyle t$	$=$	$\displaystyle 4-\frac{6}{5}$
	↓	Write 4 as a fraction with a 4 in the denominator
$\displaystyle t$	$=$	$\displaystyle \frac{20}{5}-\frac{6}{5}$
	↓	subtract
$\displaystyle t$	$=$	$\displaystyle \frac{14}{5}$
$\displaystyle t$	$=$	$\displaystyle 2.8$
	↓	Setze t in f(x) ein.

${f}({x})=\frac{4}{5}{x}+2.8$

Determining the axis intercepts

Set $f (x) = 0$ , to obtain the zeros of the function.

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

$\Rightarrow$ Intersection with the $x$ -axis at $(-\frac 7 2|0)$

The $y$ -axis intercept corresponds to the point of intersection with the $y$ -axis.

Here $t=2.8=\frac{14}{5}$

$\Rightarrow$ Intersection with the $y$ -axis at $(0|\frac {14}{5})$ .

Plot

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Determining the function equaition

Determining the yyy-axis intercept

Determining the xxx-axis intercept

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Determining the function equation

Determining the axis intercepts

Plot

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Determining the function equation

Determining the axis intercepts

Plot

Do you have a question?

Determining the function equation

Determining the axis intercepts

Plot

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Determining the $y$ -axis intercept

Determining the $x$ -axis intercept