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.