# Loese after b 7b 10 4

### Two points are enough! - Drawing solution

To determine the functional equation of a linear function, it is sufficient to know two points.

#### Example:

A straight line goes through the two points $$ A (–2 | 5) $$ and $$ B (3 | 2,5) $$. If you draw these 2 points in the coordinate system, you can determine the functional equation.

**Step 1:** Draw the two points in a coordinate system and draw the straight line with a ruler.

**Step 2:** Read off the point of intersection with the $$ y $$ axis $$ (0 | b) $$.

The $$ y $$ intercept point is $$ (0 | 4) $$.

You already know: $$ 4 $$ is the $$ y $$ value belonging to $$ x = 0 $$.

In the function equation, $$ b = 4 $$.

A straight line is determined by two points.

A linear function has a straight line as a graph.

### Drawing solution

**Step 3:** Use the slope triangle to determine the slope

$$ 2 $$ to the right, $$ 1 $$ downward → $$ m = -1 / 2 $$

**Step 4:** Set up the functional equation $$ y = f (x) = mx + b $$.

You now know m and b and can write down the functional equation:

$$ f (x) = -1/2 x + 4 $$

In the equation $$ f (x) = mx + b $$, $$ m $$ gives the slope and $$ b $$ the section on the $$ y $$ axis.

### Calculate function equation

The slope can also be calculated by taking the difference in the $$ y $$ values divided by the difference in the $$ x $$ values for the slope triangle, i.e.

$$ m = {\ text {difference between} y \ text {values}} / {\ text {difference between} x \ text {values}} $$

**Step 1:** Calculate the slope.

$$ m = {\ text {difference between} y \ text {values}} / {\ text {difference between} x \ text {values}} = {2.5-5} / (3 - (- 2 )) = - 2.5 / 5 = 1/2 $$

You now know that the function term $$ f (x) = -0.5 x + b $$, but you do not yet know the intercept $$ b $$.

**Step 2:** Insert the coordinates of the point $$ A (-2 | 5) $$ into the half-finished function equation:

$$ f (-2) = 5 $$

$$ (- 0.5) * (- 2) + b = 5 $$

**Step 3:** Solve for $$ b $$:

$$ (- 0.5) * (- 2) + b = 5 $$

$$ 1 + b = 5 $$ $$ | -1 $$

$$ b = 4 $$

**Step 4:** Write down the functional term:

$$ f (x) = -0.5 x + 4 $$

Each point on the graph can be calculated using the function equation:

$$ f (x) $$ is the $$ y $$ value for $$ x $$.

This means vice versa:

Every point on the straight line must satisfy the functional equation.

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### In the football stadium - drawing solution

There shouldn't be too much crowding after a soccer game. That is why the same number of spectators have to leave the stadium every minute.

After 10 minutes there are still 20,000 spectators in the stadium, after 15 minutes there are still 7,500.

Create a functional equation that you can use to calculate the number of viewers who have watched the game.

**Step 1:** The two points $$ A (10 | 20000) $$ and $$ B (15 | 7500) $$ result from the exercise text.

Draw them in a coordinate system.

**Step 2:** Draw the straight line with a ruler and read the intersection $$ (0 | b) $$ with the $$ y $$ axis.

The $$ y $$ axis intercept point is $$ (0 | 45000) $$. So there were 45,000 spectators.

In the function equation, $$ b = 45000 $$.

### This is how it looks in general

The associated linear function term can be calculated from the coordinates of two points $$ P_1 (x_1 | y_1) $$ and $$ P_2 (x_2 | y_2) $$:

- Calculate the slope. $$ m = (y_2-y_1) / (x_2-x_1) $$

You now know that the function term $$ f (x) = m x + b $$, you still have to calculate $$ b $$.

- Substitute the coordinates of one of the points into the half-finished function equation, e.g.

$$ f (x_1) = y_1 $$

$$ m * x_1 + b = y_1 $$

Solve for $$ b $$.

Write down the functional term:

$$ f (x) = m * x + b $$

### Drawing solution

**Step 3:** Go from the $$ y $$ axis intercept point 1 to the right and from there parallel to the $$ y $$ axis up to the straight line.

You can read off the gradient from the gradient triangle.

The slope is $$ m = - 2500. $$

**Step 4:** Set up the functional equation $$ y = f (x) = mx + b $$.

You now know $$ m $$ and $$ b $$ and can write down the functional equation:

$$ f (x) = -2500 x + 45000 $$

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### That's how it works arithmetically

Again the most important numbers:

After 10 minutes there are still 20,000 spectators in the stadium, after 15 minutes there are still 7,500.

Create a functional equation that you can use to calculate the number of viewers who have watched the game.

**Step 1:** Calculate the slope.

$$ m = {\ text {difference between} y \ text {values}} / {\ text {difference between} x \ text {values}} = (7500-20000) / (15-10) = - 12500 / 5 = -2500 $$

You now know that the function term $$ f (x) = –2500 x + b $$, but you do not yet know the intercept $$ b $$.

**Step 2:** Insert the coordinates of the point $$ A (10 | 20000) $$ into the half-finished function equation:

$$ f (10) = 20000 $$

$$ (- 2500) * (10) + b = 20000 $$

**Step 3:** Solve for $$ b $$:

$$ (- 2500) * (10) + b = 20000 $$

$$ - 25000 + b = 20000 $$ $$ | + 25000 $$

$$ b = 45000 $$

**Step 4:** Write down the function term: $$ f (x) = –2500 x + 45000 $$

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