4 ● Chapter 2 – Radical Functions
Pre-Calculus 12
Solving Radical Equations Solving with a Graphing Calculator
Example 1: Solve 64 −=− xx using a graphing calculator.
Method 1: Use of a single function.
Method 2: Use of a system of two functions.
Example 2: Solve 35 =+x algebraically. Identify any restrictions on the variables.
[ , ] [ , ]
[ , ] [ , ]
Chapter 2 – Radical Functions ● 5
Pre-Calculus 12
Example 3: Solve 392 2=+−+ xx algebraically. Identify any restrictions on the variables.
Example 4: An engineer designs a roller coaster that involves a vertical drop section just below the top of the
ride. She uses the equation advv 2)( 2
0 += to model the velocity, v, in feet per second, of the ride’s cars
after dropping a distance, d, in feet, with an initial velocity, v0, in feet per second, at the top of the drop, and
constant acceleration, a, in feet per second squared. Determine the initial velocity required in a roller coaster
design if the velocity will be 26m/s at the bottom of a vertical drop of 34m. (Acceleration due to gravity in SI
units is 9.8 m/s2).
Assignment: page 96-97 #5-7,9-11
6 ● Chapter 9 – Rational Functions
Pre-Calculus 12
Rational Functions and Transformations
Definition: A rational function is a function of the form:
)(
)()(
xh
xgxf = , where g(x) and h(x) are polynomials and h(x) ≠ 0.
Definition: Asymptotes – for a graph is a line that the graph approaches. Asymptotes are usually denoted on a graph
as a dotted line.
Examples of rational functions:
xxf
1)( = ,
3
2)(
−=
x
xxg ,
1
2)(
2
2
+=
x
xxh , 272 −+= xxy ,
27
13
2
−−
+
xx
x
Rational Functions can also transform the base rational function, x
y1
= .
khx
ay +
−=
Example 1: Graph i) x
y1
= ii) x
y10
= iii) 23
1+
+−=
xy . Identify the characteristics of the
graphs, including the behavior of the function for its non-permissible value.
Chapter 9 – Rational Functions ● 7
Pre-Calculus 12
Example 2: Given the graph, write an equation in the form khx
ay +
−= .
Rational Function in the Form of Linear Function over Linear Function
Example 3: Graph the function4
22
−
+=
x
xy . Identify any asymptotes and intercepts.
8 ● Chapter 9 – Rational Functions
Pre-Calculus 12
Comparing Rational Functions
Example 3: Consider the functions2
1)(
xxf = ,
2510
3)(
2 +−=
xxxg , and
( )24
16)(
+−=
xxh . Graph each pair
of functions.
• )(xf and )(xg
• )(xf and )(xh
Compare the characteristics of the graphs of the functions.
Assignment: 442-443 #2-7, 10