Exercises: Linear functions, zeros, axis intercepts

Checking with sketch

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

If you look at the course of 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.

Computational verification

Plug $A(1|-1)$ into $g$:

Plug the coordinates y=-1 and x=1 into the equation.

$-1=2\cdot 1-3$

That is a true statement.

⇒ $A$ is on $g$.

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

Plug $C$ into $h$:

$5=-0.5\cdot (-6)+3$

⇒ $5=3+3$

This statement is false, so $C$ is not on $h$.

Plug $D$ into $h$:

$55=-0.5\cdot (-102)+3$

⇒ $55=51+3$

This statement is false, so $D$ is not on $h$.

Plug $E$ into $g$:

$87=2\cdot 45-3$

This statement is correct, so $E$ lies on $g$.

Complete 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.

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

⇒ $P(5|\mathrm{0,5})$

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

Proof for $T$

$T(2.4|1.8)$

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

into $g$:

into $h$:

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

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