Concept:
A rational function may be constructed from information about features of its graph.
Graphing Rational Functions
vertical
x =
asymptotes
multiplicities
x-intercepts / crossing points are where the graph intersects the non-vertical asymptote.
x =
crossing points crossing points are where the graph intersects the non-vertical asymptote.
multiplicities
x =
holes
y =
x =
points
y =
oblique/horizontal asymptote
known point
slope
y-intercept
(, )
f(x) =
| + | |
INSTRUCTIONS
EXPLORATIONS
PRACTICE QUESTIONS
1. A rational function has the vertical asymptotes x = –7, x = –2, x = 3, and
x = 6, with multiplicities 2, 1, 1, and 3, respectively. It is known to pass through the point (2.8, 3.31). It has no crossing points, x-intercepts, or holes. How many of its branches lie below
the x-axis?
x = 6, with multiplicities 2, 1, 1, and 3, respectively. It is known to pass through the point (2.8, 3.31). It has no crossing points, x-intercepts, or holes. How many of its branches lie below
the x-axis?
2. A rational function f(x), whose graph doesn’t have any holes, has
x-intercepts –1 with multiplicity 1 and 2 with multiplicity 2, and a y-intercept
of –0.2. It also has the vertical asymptotes x = –4 with multiplicity 1,
x = 1 with multiplicity 1, and x = 5 with multiplicity 2. What is f(3.5)?
x-intercepts –1 with multiplicity 1 and 2 with multiplicity 2, and a y-intercept
of –0.2. It also has the vertical asymptotes x = –4 with multiplicity 1,
x = 1 with multiplicity 1, and x = 5 with multiplicity 2. What is f(3.5)?
3. A rational function with horizontal asymptote y = 3 contains the point
(4.4, –7). It has crossing points at –2 and 7, both with multiplicity 2, and vertical asymptotes x = –6 with multiplicity 2 and x = 4 with multiplicity 3. There is a hole where x = 3.2. What is the y-value of the hole?
(4.4, –7). It has crossing points at –2 and 7, both with multiplicity 2, and vertical asymptotes x = –6 with multiplicity 2 and x = 4 with multiplicity 3. There is a hole where x = 3.2. What is the y-value of the hole?
4. The line y = –0.25x + 1 is an oblique asymptote for a rational function f(x) whose crossing points are –5 and 6, both with multiplicity 1, and vertical asymptotes x = –8, x = –1, and x = 3, all with multiplicity 1. It is known that
f(–9) = 2. For which other x-value is
f(x) = 2?
f(–9) = 2. For which other x-value is
f(x) = 2?