📚 Practice With Worksheets
Practice finding slope from graphs, connecting linear representations, and writing equations in slope-intercept form.
▶️ Learn With Videos
Review rate of change, graphing with \(y=mx+b\), and finding the equation of a line from its graph.
🛠️ How to Use This Tool
Enter a linear equation in slope-intercept form, or enter separate values for slope \(m\) and y-intercept \(b\). Fractions and decimals may be used.
Turn the rise-and-run stairs and labels on or off to connect the graph to the slope. The stair moves vertically by the rise first and then horizontally by the positive run.
Drag the coordinate plane to pan, pinch or use the mouse wheel to zoom, or use the Zoom In, Zoom Out, and Reset View buttons. Hide the controls when more graphing space is helpful.
🧠 Things to Notice
A positive slope rises from left to right, while a negative slope falls. A larger value of \(|m|\) creates a steeper line, and a value closer to zero creates a shallower line.
The rise may be positive, negative, or zero, but the run is shown as a positive value. For example, \(m=-\frac{3}{4}\) means rise \(-3\), then run \(4\).
Changing \(b\) moves the line vertically without changing its slope. Lines with the same slope and different y-intercepts are parallel.
📘 Vocabulary
Slope: A measure of a line’s direction and steepness, calculated as vertical change divided by horizontal change.
Rise: The vertical change between two points on a line.
Run: The horizontal change between two points on a line.
Rate of Change: The amount the output changes for each one-unit change in the input.
y-Intercept: The point where a graph crosses the y-axis. Its x-coordinate is always \(0\).
x-Intercept: The point where a graph crosses the x-axis. Its y-coordinate is always \(0\).
Slope-Intercept Form: The form \(y=mx+b\), where \(m\) is the slope and \(b\) is the y-intercept.
Standard Form: A linear equation written as \(Ax+By=C\).
Positive Slope: A slope that makes a line rise from left to right.
Negative Slope: A slope that makes a line fall from left to right.
Zero Slope: The slope of a horizontal line.
Undefined Slope: The slope of a vertical line because its horizontal change is zero.
📐 Slope and Linear Equation Concepts
Slope can be expressed as rise over run, change in \(y\) over change in \(x\), or the difference between the y-coordinates divided by the difference between the x-coordinates.
\(m=\dfrac{\text{rise}}{\text{run}}=\dfrac{\Delta y}{\Delta x}=\dfrac{y_2-y_1}{x_2-x_1}\)
Slope-intercept form displays the slope and y-intercept directly. Standard form can be rearranged into slope-intercept form when the line is not vertical.
\(y=mx+b \qquad Ax+By=C\)
At the y-intercept, \(x=0\). At the x-intercept, \(y=0\). Substituting zero for the appropriate variable helps locate each intercept.
\(\text{y-intercept: }x=0 \qquad \text{x-intercept: }y=0\)
Horizontal lines have slope \(0\), equations of the form \(y=b\), and y-intercept \((0,b)\). Vertical lines have undefined slope and equations of the form \(x=c\).
\(y=b \Rightarrow m=0 \qquad x=c \Rightarrow m\text{ is undefined}\)
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