graph of a linear function - Der Graph einer linearen Funktion

Adjust the gradient (slope) and the y-intercept with the sliders. Mit den Schiebereglern kannst du die Steigung und den y-Abschnitt verändern.

$$y = mx + c$$

source: jsxgraph.org/wiki/index.php/Slider_and_function_plot

the gradient (slope) - Die Steigung

The slope (gradient) is the factor that determines the steepness of the line graph.

m > 0 - a positive gradient

as the x-values increase the y-values increase

m < 0 - a negative gradient

as the x-values increase the y-values decrease

m = 0

If the gradient is equal to 0, the line is a horizontal line passing through the y-intercept.

example

$${y = 3}$$ or $${f(x) = 7}$$

the y-intercept - Der y-Abschnitt

the y-intercept is the value of y where the graph of the line intersects the y-axis. Note that on the y-axis, the x-value is 0.

example

$${y = x - 2}$$ substituting $${x = 0}$$ y is then equal to $${y = 0 - 2 = -2}$$ the y-intercept is -2

find the function f(x) = mx + c (linear equation)

Some steps to different problems in a GeoGebra book

given the slope and the y-intercept

As m is the slope (gradient) and c is the y-intercept, you only need to substitute these in the equation of a linear function.

$$y = mx + c$$

given the y-intercept and another point of the line

Here, the slope m is missing.

• Substitute the y-intercept c into the equation. Take the x- and y-coordinate of the point and also substitute these into the equation.
• Now, solve for m to find the slope (gradient, Steigung)

given two points

Given two points $P(x_1,y_1)$ and $Q(x_2,y_2)$. The method is

• find the gradient with $m =\frac{y_2 - y_1}{x_2 - x_1}$ (Note - it does not matter which point you give the index 1 or 2)
• substitute the coordinates of one of the points to find the y-intercept c.
• write the equation of the line

function notation f(x)

So far you have graphed using x and y. Instead of using y we now call it f(x) - which reads “f of x”. This is a name for the expression that gives you a rule to use x with and create a new value f(x). When you want to graph it, f(x) is set equal to the y (you have used before)

$$f(x) = mx + c$$

examples

$$f(x) = 2x + 3$$

$$g(x) = -\frac{2}{3}x - 6$$

Note - instead of f you can use any other letter (usually g and h). But, it could also be V(x) for volume, A(x) for area or s(t) - the distance s depending on time t.