Outlines with graphs and examples how to find the equation of a line by reading on the graph, using it to calculate special points, finding it with point and slope, and using two points.

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In these explanations you will learn how to set up functional equations for linear functions.

- How to find the equation of a line by reading on the graph
- How to find the equation of a line to calculate special points
- How to find the equation of a line with point and slope
- How to find the equation of a line using two points

### How to find the equation of a line by reading on the graph

##### The graph of a linear function is a straight line. The equation has the form y = m x + by=mx+b.

Here is designated m the value for the slope and b the y-Axis section.

If you have given the graph, i.e. the straight line, of a linear function, you can take both values directly from the graph.

Example 1: Determine the function equation for the graph shown.

Solution:

- The straight line intersects the y-axis at the point -4… so b = −4
- On the slope triangle below you can see that the straight line has the slope m = 3
- So:
**y = 3 x – 4**

Example 2: Determine the function equation for the graph shown

- The straight line intersects the y-axis at point 1, so b = 1
- At the slope triangle below you can see that the straight line has the slope m = -2/3.
- So …
**y = -2/3x + 1**

### How to find the equation of a line to calculate special points

Many relationships in nature and technology can be described using linear functions.

Often the special points of the graphs (zero point, y-axis intercept, points of intersection with other graphs) also have a special meaning in these examples.In one region, a certain type of plant is infected with a virus. The percentage of sick plants is recorded and registered every 5 years.

A linear function describes the trend quite well, revealing the proportion of diseased plants decreases and the plants become immune to the virus. Be at the first count 880% of plants counted as sick.What proportion of the plants will still be sick after 70 years?When will there be no more sick plants in this region for the first time?

Establish functional equation

You can determine the slope of the straight line with the help of a slope triangle:

- m = -30/35 = -6/7
- The straight line intersects the y-axis at the point b = 80.
- At the first count 80% of all plants were sick.
- y = -6/7 x + 80

You plug the value 70 for x into the function equation and calculate y.

f (70) = 20

Calculate zero

The calculated point in time at which no more sick plants are expected to be counted corresponds to the zero point of the function.

You calculate the zero (f (x) = 0) by inserting 0 for y into the function equation and solving the equation for x.

Since the census takes place every 5 years, one will probably not find any more diseased plants for the first time after 95 years. x = 93 1/3

### How to find the equation of a line with point and slope

If you have given the slope and a point of the graph of the function, you can calculate the y-axis intercept.The graph of a linear function f passes through the point P ( 2 | -5 ) and has the slope m = -3/2.

Find the functional equation of f.

The point Q ( −2-| y) is also on the graph of the function. Find the y-coordinate.

The point P ( 2 | -5 ) lies on the straight. So its coordinates satisfy the corresponding functional equation.The functional equation has the form y = -3/2x + b

You determine the y-axis intercept b by inserting the coordinates of the point P into the function equation and solving the equation for b:

y = -3/2 x – 2

Determine the y-coordinate of the point Q.

You insert the x-coordinate of the point Q into the function equation and calculate y.

y = 1 Q ( −2 | 1 )

### How to find it using two points

If you have given two points P and Q of the graph of the function, you can calculate the slope with the help of the slope formula m =yq-yp/xq-xp.

With the help of the function equation you can then check whether any other point is also on the graph of the function.Determine the functional equation for the straight line that passes through the points P ( −2 |22 ) and Q (-4 | −1 ) runs.

Check whether the points R ( −4 | 3 ) and S ( 1 | 1 ) lie on the straight.

Calculate the slope using the slope formula

You insert the coordinates of the points P and Q into the slope formula and calculate the slope.

Determine the y-axis intercept and state the function equation

You put the coordinates of one of the two points in the function equation y = -12x + b and solve the equation for b. With P ( −2 | 2 ) it applies thus:

- b = 1
- y = -1/2 x + 1

Check that point R belongs

You insert the coordinates of the point R into the function equation and check whether a true statement arises.

The point R ( −4 | 3 ) lies on the straight.

Check whether point S belongs

You insert the coordinates of the point S into the function equation and check whether a true statement arises.

The point S ( 1 | 1 ) is not on the straight.