Properties and Graph of Linear Functions

In the previous three posts, we have learned about graph of linear functions in relations to its graph. We summarize these learnings in this posts.

The linear function has equation y = ax + b where a and b are real numbers. The number a is the slope of the graph of the function and the number b is the y-intercept. The sign of a determines the direction of the graph (click here for detailed explanation). If a > 0, the function is increasing and if a < 0 then the function is decreasing. If a = 0, the graph of the function is a horizontal line.

The value of a determines the steepness of the graph (click here for detailed explanation). As the absolute value of a increases, the graph becomes steeper. The value of b increases the value of the function by b if b > 0 and decreases the value of the function if b < 0 (click here for a detailed explanation). Graphically, this translates the function vertically — up of b > 0, down if b < 0.

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The effect of the sign of the slope in y = ax

A linear function is a function whose equation is of the form y = ax + b. We separate the discussion about it into two parts: b = 0 and b \neq 0. In this post, we only discuss the graph of y = ax + b where b = 0. We discuss the effect of the sign of the slope in y = ax.

If we let b = 0, the equation y = ax + b becomes y = ax + 0 or simply y = ax.

Notice that if x = 0, then y = ax = a(0) = 0. This means that the graph contains the point with coordinates (0,0). Therefore, y = ax passes through the origin.

Generalization 1: The graph y = ax passes through the origin.

We now examine the effect of the values of a. There are three cases: a = 0, a > 0, and a < 0» Read more

The Curve Sketching Series

This series discusses the strategies on graphing different functions particularly linear, quadratic, and rational functions.

Curve Sketching 1 is a discussion of the four strategies in graphing linear functions. This includes two points, slope and intercept, translation, and x and y intercepts.

Curve Sketching 2 is a discussion about sketching the graph of quadratic functions. To be able to graph this function you need its critical points such as maximum or minimum, x intercepts, and y-intercept. It is also important to know where the graph opens and the axis of symmetry.

Curve Sketching 3 is a discussion about the vertical and horizontal asymptotes of rational functions. The vertical asymptote of a function is what makes f(x) = n/0 and the horizontal asymptote is the quotient of the leading terms if they have the same power.


Curve Sketching 4 is a discussion about the oblique asymptote of rational functions. An oblique asymptote exists if the degree of the numerator is 1 more than the degree of the denominator.

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