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Math Lesson 14.1.4 - Steep Linear Graphs
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Welcome to our Math lesson on Steep Linear Graphs, this is the fourth lesson of our suite of math lessons covering the topic of Linear Graphs, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
Steep Linear Graphs
If a graph line is steep, this means that none of the variables are zero because both variables change when shifting along the graph. We can identify a steep graph by looking at the coefficient a and b. If both of them are different from zero, the graph is steep. The general formula of such graphs in two dimensions is
For example, the line
has a steep graph, as none of the coefficients are zero (a = 2 and b = 5). This condition does not include the constant c in the sense that it does not matter what the value of c is; the steepness of the graph does not depend on it. To prove this, we can show two lines in the same graph:
where the coefficient a and b are the same on both lines while the constant c is different (in the first line c = -1 and in the second line c = 0). Look at the figure below that shows the graphs of the two lines above.
As you see, the two graphs are parallel; the only thing that makes them different is a vertical shift of one graph in respect to the other, which comes from the fact that the constant c is different. From here, we can conclude that the constant c plays a role in the vertical shift of a linear graph (but not only). On the other hand, the coefficient a are equal in two parallel lines; so are the coefficients b as well.
We can therefore calculate the value of the constant c by taking x = 0 and solving the rest of the equation
If the coefficient b is not provided, we can see what the y-intercept of the line is, as it corresponds to the value of the constant c.
Example 3
The graphs of the lines -3x + 2y - 5 = 0 and ax + by + 1 = 0 are parallel. What are the values of a and b? Plot the two lines in the same coordinates system.
Solution 3
We explained earlier using theory that if the corresponding coefficients of two lines are equal, these lines are parallel. Hence, for the coefficient a and b of the second line we have: a = -3 and b = 2. This means the second line has the equation -3x + 2y + 1 = 0.
The graphs of the two lines are parallel given their equal corresponding coefficients. They are shown in the figure below.
More Linear Graphs Lessons and Learning Resources
Whats next?
Enjoy the "Steep Linear Graphs" math lesson? People who liked the "Linear Graphs lesson found the following resources useful:
- Steep Feedback. Helps other - Leave a rating for this steep (see below)
- Linear Graphs Math tutorial: Linear Graphs. Read the Linear Graphs math tutorial and build your math knowledge of Linear Graphs
- Linear Graphs Revision Notes: Linear Graphs. Print the notes so you can revise the key points covered in the math tutorial for Linear Graphs
- Linear Graphs Practice Questions: Linear Graphs. Test and improve your knowledge of Linear Graphs with example questins and answers
- Check your calculations for Linear Graphs questions with our excellent Linear Graphs calculators which contain full equations and calculations clearly displayed line by line. See the Linear Graphs Calculators by iCalculator™ below.
- Continuing learning linear graphs - read our next math tutorial: Slopes and Gradients
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