Exercises: Linear functions, zeros, axis intercepts

Part 1

Drawing a sketch

Select any point on the straight line, e.g. the $y$-axis intercepts $(0|-3)$ and $(0|3)$. Go from there 1 unit to the right and, corresponding to the slopes, $m_g=2$ upwards and $m_h=-0.5$ downwards. Connect the two points to form a straight line.

If you look at the straight line and, for example, the position of point $A$, you will see that it will hardly lie on the straight line $h$, but probably on $g$. Similarly, you can decide for other points whether a mathematical check is worthwhile: Point $D(-102/55)$ can only lie on $h$, for example.

Check by computation

$A(1|-1)$ on $g$

Put the coordinates of the points into the equation in question. So set $y=-1$ and $x=1$.

$-1=2\cdot 1-3$

Check if the equation is a true statement.

$-1=-1$

Thi is a true statement.

$\Rightarrow$$A$ lies on $g$.

$B$ does not lie on any of the straight lines. This can be clearly seen in the sketch.

$C$ is plugged into $h$: $5=-\mathrm{0,5}\cdot (-6)+3\Rightarrow 5=3+3$ This statement is false, so $C$ does not lie on $h$.

$D$ is plugged into $h$: $55=-0.5\cdot (-102)+3\Rightarrow 55=51+3$ This statement is false, so $D$ does not lie on $h$.

$E$ is plugged into $g$: $87=2\cdot 45-3$ This statement is true, so $E$ lies on $g$.

Part 2

Completing coordinates

$h:y=-0.5x+3$ ; $P(5|?)$

The given coordinate of the point (the $x$-coordinate) is inserted into the function equation and the missing $y$-coordinate is calculated from it.

$y=-0.5\cdot 5+3=-2.5+3=0.5$

$\Rightarrow$$P(5|\mathrm{0,5})$

Q: $y=-0.5\cdot (-3.5)+3=1.75+3=4.75$ $\Rightarrow$ $Q(-3.5|4.75)$

R: $12=-0.5x+3\Rightarrow 0.5x=-9\Rightarrow x=-18$ $\Rightarrow$ $R(-18|12)$

S: $-7.5=-0.5x+3\Rightarrow 0.5x=10.5\Rightarrow x=21$ $\Rightarrow$ $S(21|-7.5)$

Part 3

Proof for $T$

$T(2.4|1.8)$

Insert the coordinates of $T$ into both line equations. If the statements are true, $T$ lies on the lines.

on $g$: $1.8=2\cdot 2.4-3$

$1.8=4.8-3$

$1.8=1.8$

on $h$: $1.8=-0.5\cdot 2.4+3$

$1.8=-1.2+3$

$1.8=1.8$

$\Rightarrow$Both equations give correct statements, so the point $T$ lies on both straight lines.

$\Rightarrow$$T$ Is the intersection point of the lines.