Applied exercises: Linear functions

Jonathan and Hannes decide to climb a very high mountain in the alps. Jonathan starts his hike at sea level (0m), while Hannes starts in a small shack at an altitude of (2500m).

Both may climb at 500m per hour. The function term describing Jonathan's ascent is

$j (x) = 500 x$

with $x$ denoting the time in hours. Decide which function term describes Hannes' ascent!

$h (x) = 500 x + 2500$
$h (x) = 2500 x$
$h (x) = 500 x - 2500$

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Applied exercises: Linear functions

Ascent of Hannes

To find the correct function term, you have to find the slope and the $y$ -axis intercept, given the values from the problem setting.

Since Jonathan and Hannes climb at the same speed, the function term of Hannes' ascent must have the same slope as the function term of Jonathan's ascent.

Jonathan's ascent is described with the function term $j (x) = 500 x$ .

Therefore, the slope of $j$ is 500.

The possible term $h (x) = 2500 x$ for Hannes' ascent is thereforeruled out. It would have a wrong slope!

Now two function terms with the correct slope but different $y$ -axis intercepts remain. Which of the terms is correct?

Compare the two $y$ -axis intercepts. These are $+ 2500$ and $- 2500$ .

You can quickly see that $+ 2500$ must be correct, because Hannes starts above Jonathan and therefore must be at the beginning $(x = 0)$ at $h (0) = 2500$ . By contrast, $h (0) = - 2500$ would mean that Hannes would start BELOW sea level, which does not make sense!

So you get a slope of 500 and a y-axis intercept of 2500.

The correct function term for Hannes' ascent is hence $h (x) = 500 x + 2500$ .

Applied exercises: Linear functions

Ascent of Hannes

To find the correct function term, you have to find the slope and the y-axis intercept, given the values from the problem setting.

Since Jonathan and Hannes climb at the same speed, the function term of Hannes' ascent must have the same slope as the function term of Jonathan's ascent.

Jonathan's ascent is described with the function term j(x)=500x.

Therefore, the slope of j is 500.

The possible term h(x)=2500x for Hannes' ascent is thereforeruled out. It would have a wrong slope!

Now two function terms with the correct slope but different y-axis intercepts remain. Which of the terms is correct?

Compare the two y-axis intercepts. These are +2500 and −2500.

You can quickly see that +2500 must be correct, because Hannes starts above Jonathan and therefore must be at the beginning (x=0) at h(0)=2500 . By contrast, h(0)=−2500 would mean that Hannes would start BELOW sea level, which does not make sense!

So you get a slope of 500 and a y-axis intercept of 2500.

The correct function term for Hannes' ascent is hence h(x)=500x+2500.

To find the correct function term, you have to find the slope and the $y$ -axis intercept, given the values from the problem setting.

Jonathan's ascent is described with the function term $j (x) = 500 x$ .

Therefore, the slope of $j$ is 500.

The possible term $h (x) = 2500 x$ for Hannes' ascent is thereforeruled out. It would have a wrong slope!

Now two function terms with the correct slope but different $y$ -axis intercepts remain. Which of the terms is correct?

Compare the two $y$ -axis intercepts. These are $+ 2500$ and $- 2500$ .

You can quickly see that $+ 2500$ must be correct, because Hannes starts above Jonathan and therefore must be at the beginning $(x = 0)$ at $h (0) = 2500$ . By contrast, $h (0) = - 2500$ would mean that Hannes would start BELOW sea level, which does not make sense!

The correct function term for Hannes' ascent is hence $h (x) = 500 x + 2500$ .