Describing Key Features of Functions

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In addition to interpreting functions in context, it can be worthwhile to analyze the graphs of functions in terms of key features, which include the following.

  • xx- and yy-intercepts
  • increasing/decreasing intervals
  • relative minimum/maximum values
  • symmetries
  • end behavior
Concept

Intercept

The points where a graph crosses the x-x\text{-} and y-y\text{-}axes are called the x-x\text{-}intercept and y-y\text{-}intercept, respectively.

Sometimes, only one coordinate of these points is referenced. For example, since the x-x\text{-}intercept lies at (a,0),(a,0), it can be said that x=a.x=a. The same is true for the y-y\text{-}intercept (0,b),(0,b), y=b.y=b. A relation can have several intercepts; a function can have multiple x-x\text{-}intercepts, but it can only have one y-y\text{-}intercept.
Concept

Increasing and Decreasing Intervals

A function is said to be increasing when, as the xx-values increase (from left to right), the values of f(x)f(x) increase. Conversely, a function is said to be decreasing when, as xx increases, f(x)f(x) decreases. The graph below shows increasing intervals with green arrows and decreasing intervals with red arrows.

The function above contains two increasing intervals and one decreasing interval. To describe each, use the xx-values. Commonly a point where a function has a relative maximum or a relative minimum is neither included in an increasing nor a decreasing interval. It is necessary to scan the graph from left to right. From left side of graph to x=-2increasingFrom x=-2 to x=0decreasingFrom x=0 to right side of graphincreasing\begin{aligned} \text{From left side of graph to} \ x=\text{-} 2 & \quad \rightarrow && \text{increasing} \\ \text{From} \ x=\text{-} 2 \ \text{to} \ x=0 & \quad \rightarrow && \text{decreasing} \\ \text{From} \ x=0 \ \text{to right side of graph} & \quad \rightarrow && \text{increasing} \\ \end{aligned} Although the entire graph cannot be shown, it is reasonable to assume it continues in the same manner. Thus, for all xx-values less than x=-2,x=\text{-} 2, ff will be increasing. Additionally, for all xx-values greater than x=0,x=0, ff will be increasing. Thus, the above intervals can be expressed as follows.

Increasing interval: -<x<-2 and 0<x<+Decreasing interval: -2<x<0\begin{aligned} \textbf{Increasing interval:} & \ \text{-} \infty < x < \text{-} 2 \ \text{and} \ 0 < x < + \infty \\[0.8em] \textbf{Decreasing interval:} & \qquad \qquad \ \text{-} 2 < x < 0 \\ \end{aligned}
Concept

Relative Minimum and Maximum

For some functions, their graphs extend infinitely in the vertical direction. For these graphs, there are no highest or lowest points. However, it's possible for these functions to have relative minimum or relative maximum values. A relative minimum is the lowest point for a region of the graph. Similarly, a relative maximum is the highest point for a region of the graph.

Notice that (-2,3)(\text{-} 2,3) is the highest point on the graph for all xx-values less than 1.1. Thus, (-2,3)(\text{-} 2,3) is a relative maximum. Notice also that (0,-1)(0, \text{-} 1) is the lowest point on the graph for all xx-values greater than -3.\text{-} 3. Thus, (0,-1)(0,\text{-} 1) is a relative minimum.
Concept

Symmetry

There are two types of symmetry that the graph of a function can have — even or odd. A function has even symmetry if it is symmetric with respect to the yy-axis. In other words, if the yy-axis cuts the graph into two mirror images.

Notice that if the graph were folded vertically on the yy-axis, the marked points would lie on top of each other. This is true for every point on f.f. Thus, f(x)f(x) has even symmetry. A function is said to have odd symmetry if it's symmetric about the origin. In other words, if one half of the graph can be rotated 180180^\circ to match the other half of the graph exactly.

Notice that the portion of the graph below the yy-axis could be rotated so that it lies directly on top of the portion above the yy-axis. Thus, ff has odd symmetry.
Concept

End Behavior

The end behavior of a function is the direction toward which f(x)f(x) moves as xx extends to the left and the right infinitely. Analyzing each end individually, if f(x)f(x) extends upward, the behavior is expressed as f(x)+. f(x) \rightarrow + \infty. Alternatively, if f(x)f(x) extends downward, the behavior is f(x)-. f(x) \rightarrow \text{-} \infty. Consider the function f(x).f(x).

From the arrows on the graph, it can be seen that the left-end extends downward, while the right-end extends upward. Thus, the end behavior of ff can be expressed as follows. As x-f(x)-As x+f(x)+\begin{aligned} \text{As} \ x \rightarrow \text{-} \infty &\text{,} \ &&f(x) \rightarrow \text{-} \infty \\ \text{As} \ x \rightarrow + \infty &\text{,} \ &&f(x) \rightarrow + \infty \\ \end{aligned}

A polynomial function with a term with the highest degree of axn,ax^n, where aa is the leading coefficient and nn is the degree of the polynomial the end behavior depends on aa and n.n.

a>0, \textcolor{blue}{a}>0,\ n even{\color{#FF0000}{n}}\ \text{even}

a>0, \textcolor{blue}{a}>0,\ n odd{\color{#FF0000}{n}}\ \text{odd}

a<0, \textcolor{blue}{a}<0,\ n even{\color{#FF0000}{n}}\ \text{even}

a<0, \textcolor{blue}{a}<0,\ n odd{\color{#FF0000}{n}}\ \text{odd}

Reset\text{Reset}

Exercises

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