Graphs and functions are critical, not only for solving math problems, but for real life situations. They can be used, for example, to find trends in data. None of this is possible, however, without first knowing the basic foundation of graphing, the different forms that an equation can be written in, or how to write these equations. These basics will be used for all types of more complex graphing in the future.
Slope: A ratio of the distance moved vertically over the distance moved horizontally in a non-vertical line.
x-Intercept: The point where a line crosses the x-axis.
y-Intercept: The point where a line crosses the y-axis
Slope-Intercept Form: A form of writing a linear equation in two variables: y = mx+b, where m is the slope, b is they-intercept, and x and y are the variables.
An equation in two variables can be graphed on a coordinate plane.
The graph is the set of points that are solutions to the equation (they make the equation true).
If the graph is a straight line, the equation is linear.
The slope between the points \((x_1, y_1)\) and \((x_2, y_2)\)is:
\(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta y}{\Delta x}\)
\(\Delta\) is the Greek letter delta that means change
The slope \(m\) of this graph is \(5/3\).
Think of the slope as describing the steepness of the line. The slope can also represent a rate of change when one quantity is compared to another. Example: miles per hour
Two points can be used to determine a line. Two convenient points to use are the x-intercept and the y-intercept.
The slope-intercept form is a common form of writing a linear equation: y = mx+b
\(y = 3x + 2\) with \(y = mx + b\)
To graph an equation in the slope-intercept form,
Converting from standard to slope-intercept form:
\(Ax + By = C \rightarrow y = -\frac{A}{B}x + \frac{C}{B}\)
Converting from slope-intercept to point-slope form:
Converting from point-slope to slope-intercept form: