7.5 Graphing Function Families, Asymptotes & End Behavior

Graphing Function Families, Asymptotes & End Behavior

Quick Summary: The blueprint asks you to "graph linear, quadratic, square root, cube root, piecewise, absolute value, polynomial, rational, logarithmic, and exponential functions" and to "identify any intercepts, minima, maxima, asymptotes, and end behavior." Lines and parabolas were covered in Chapters 5 and 7.3. This section handles the remaining families. On a multiple-choice test you are almost never asked to plot one by hand — you are asked to match an equation to a picture or read a feature off a graph, so recognizing shapes is the skill that scores.

The Parent Function Catalogue

Telling the Look-Alikes Apart

• Square root vs. cube root. A square root graph stops — it has a visible endpoint and only exists on one side. A cube root graph continues in both directions through its center. If the picture extends to the left of its turning point, it is a cube root.
• Absolute value vs. quadratic. Both are U-ish, but $\lvert x \rvert$ has a sharp corner and straight sides; $x^2$ is smoothly curved.
• Exponential vs. logarithmic. They are reflections of one another across $y = x$. The exponential flattens toward a horizontal asymptote; the logarithm plunges toward a vertical one.

Piecewise Functions

A piecewise function applies different rules on different intervals, written with a brace and conditions:

$$f(x) = \begin{cases} 2x + 1 & x < 1 \\ x^2 & x \ge 1 \end{cases}$$

To evaluate, first decide which interval the input falls in, then use only that rule.

• $f(-2)$: since $-2 < 1$, use $2x + 1 = 2(-2) + 1 = -3$.
• $f(4)$: since $4 \ge 1$, use $x^2 = 16$.
• $f(1)$: since $1 \ge 1$ (the condition includes equality), use $x^2 = 1$.

The value at a boundary is decided entirely by which piece carries the $\le$ or $\ge$. Graphically this shows as a closed dot on the piece that includes the endpoint and an open dot on the piece that excludes it. Real-world piecewise models include tiered utility rates, tax brackets, and shipping-cost tables.

Absolute value is itself piecewise: $\lvert x \rvert$ equals $x$ when $x \ge 0$ and $-x$ when $x < 0$, which is exactly why its graph is two straight rays meeting at a corner.

Asymptotes in Rational Functions

An asymptote is a line the graph approaches without ever reaching.

Vertical Asymptotes

Set the denominator equal to zero and solve; each value where the numerator is not also zero gives a vertical asymptote.

For $f(x) = \dfrac{x + 1}{x - 3}$, the denominator is zero at $x = 3$, so $x = 3$ is a vertical asymptote. The graph shoots toward $\pm\infty$ on either side of it.

Caution — the hole. If a factor cancels between numerator and denominator, you get a hole, not an asymptote. In $f(x) = \dfrac{(x-2)(x+1)}{x-2}$ the graph is the line $y = x + 1$ with a single point removed at $x = 2$.

Horizontal Asymptotes

Compare the degrees of the numerator ($n$) and denominator ($m$):

Asymptotes in Exponential and Logarithmic Graphs

• $f(x) = 2^x$ has a horizontal asymptote at $y = 0$: as $x \to -\infty$ the output shrinks toward zero but never reaches it. This is why exponential decay models never quite hit zero.
• $f(x) = \log x$ has a vertical asymptote at $x = 0$: you cannot take the logarithm of zero or a negative number, and as $x$ approaches 0 from the right the output plunges toward $-\infty$.

End Behavior

End behavior answers: what does $f(x)$ do as $x \to +\infty$ and as $x \to -\infty$? For polynomials, only the degree and the sign of the leading coefficient matter.

So $f(x) = -2x^4 + 7x^3 - x$ has even degree and a negative lead: both ends fall. The middle terms change the wiggles, never the ends.

Reading Key Features Off Any Graph

Whatever the family, the test asks the same short list of questions:

• $x$-intercepts (zeros): where the curve crosses the $x$-axis; solve $f(x) = 0$.
• $y$-intercept: where it crosses the $y$-axis; evaluate $f(0)$.
• Relative maximum / minimum: a peak or valley — higher or lower than nearby points, though not necessarily the highest or lowest overall.
• Increasing / decreasing intervals: read left to right; the graph rises or falls.
• Positive / negative intervals: where the curve sits above or below the $x$-axis — not the same as increasing or decreasing.
• Domain and range: the set of usable inputs and the set of achieved outputs.

The most common confusion is "increasing" versus "positive." A graph can be below the axis (negative) while climbing (increasing) — for example $y = x + 3$ on the interval $-5 < x < -3$.