Unit 2a, Cluster 4: Linear and Exponential Relationships
Interactive Study Guide: Learning Cycle A
I can accurately graph a linear function by hand by identifying key features of
the function such as the x and y intercept and slope.♦
LINEAR FUNCTIONS
Review of Graphing:
Slope-Intercept Form y = _____________
Example: y = -2x + 3
Standard Form _________= C
Example: 4x – 3y = 12
Skill-based Task
1.
Graph the function f(x) = 2x – 3.
2. Graph the function f(x) = 2x
Problem Task
1.
The population of salmon in a lake triples
each year. The current population is 472.
Model the situation graphically. Include
the last three years and the next two.
Model the situation with a function.
Unit 2a, Cluster 4: Linear and Exponential Relationships
Interactive Study Guide: Learning Cycle B
I can graph a linear or exponential function using technology.
I can sketch the graph of an exponential function accurately identifying x and
y intercepts and asymptotes.
I can describe the end behavior of an exponential function (what happens as x
goes to positive or negative infinity).
EXPONENTIAL FUNCTIONS
______________ Function: f(x) = bx where b > 0, b ≠ 2 and x is any real number.
Let's examine the function f (x) = 2x. Check out the table and the graph.
Most exponential graphs will have this same __________.
This graph is very, very small on its left side and is extremely close to the x-axis.
We describe this _____ _______ by saying that as the domain goes to negative infinity the
function _____________ ______.
As the graph progresses to the right, it starts to grow faster and faster and shoots off the top of
the graph very quickly, as seen in the graph.
So describing the end behavior on the right we would say that as the domain goes to positive
infinity, the function goes to ____________ ______________.
In a linear function, the "rate of change" remains the same across the graph.
In exponential graphs, the "rate of change" ___________ or ______________ across the graphs.
Characteristics:
Exponential graphs of the form f (x) = bx have certain characteristics in common.
• graph crosses the y-axis at (___, ___)
• when b > 1, the graph __________
• when 0 < b < 1, the graph _____________
• the domain is all ________ numbers
• the range is all _________ real numbers (never ____)
• graph is ____________ to the x-axis – gets very, very close to the x-axis
but does not touch it or cross it.
Unit 2a, Cluster 4: Linear and Exponential Relationships
Interactive Study Guide: Learning Cycle C
I can discuss and compare two different functions (linear and/or exponential)
represented in different ways (tables, graphs or equations). Discussion and
comparisons should include: identifying differences in rates of change,
intercepts, and/or where each function is greater or less than the other.
COMPARING LINEAR AND/OR EXPONENTIAL FUNCTIONS
Graph
linear
y = _______
slope of m =_
___________
function
y = _______
therefore,
y = ___
(hence b=__)
linear
y=
_________
slope of m __
linear
___________
y = ________
linear
y = ________
slope of m___
linear
___________
y = ________
exponential
_________
y = _______
b _____
*growth
factor b
y =________
Exponential
__________
y =________
___________
*decay factor
b
y = ________
Rate of
Change
Table
x-int.
y-int.
Domain Range
x values y values
In a linear equation mx represents additive change; (b + m + m + m + m + ……….)
• the slope m is the constant rate of change between any two points on the linear graph/table
• (x1, y1) and (x2, y2) =
*In an exponential equation, bx represents multiplicative change; (a • b • b • b • b • ……….)
• Growth factor b = 100% + growth rate (percent)
Example: Value of savings account is growing by 2.5% per year, the growth fact is 1.025.
y = balance (1.025)x
• Decay factor b = 100% – decay rate (percent)
Example: Value of the dollar is declining each year by 3%, the decay factor is 97%.
y = 1(.97)x
• The average rate of change of points in an exponential graph/table will not be constant
• The average rate of change between any two points is the slope of the secant line between
two points (linear or non-linear).
Slope of secant line is taught in Secondary Mathematics 1 (Honors).
You should be fluent in moving between the different representations and should be able to
identify characteristics of an exponential function and linear function whether the representation
is a table, an algebraic equation, a graph, a verbal description or a real-world context.
Example:
Discuss and compare the
following
functions: f(x) = 3/2x + 5
and g(x) = 3x +1
.
Here we are comparing a _______ function, f(x) with an ___________ function, g(x).
Notice that in the linear equation the variable, x, is multiplied by the coefficient, but in the
exponential equation the x is the exponent.
So with the exponential function, the inputs become the _______.
Both functions have a constant added on the end.
In f(x), the constant, 5, tells us where the y-intercept of the graph will end up.
In g(x), the constant, 1, tells us where that the __________ asymptote of the function will be
y = 1.
These facts can both be verified by examining the graphs of both functions.
Looking at the tables for the two functions, we can examine the ________ ____ ___________
for each function.
For f(x), the rate of change stays the same – it is 3/2.
In contrast, g(x) has a rate of change that _____ _____ ________ ________.
The rate of change there is being multiplied by a factor of 3 each time.
The rates of change are both increasing, but g(x) is increasing at a __________ rate than f(x).
Going back to the graphs of the functions, we can see that f(x) has an x-intercept at (___,___)
and a y-intercept at (___, ___).
On the other hand, g(x) does _____ have an x-intercept.
The graph of g(x) ________ crosses the x-axis, but it does cross the y-axis at (___, ___).
This tells us that if we want an output of ___ from f(x) then we can input ______.
For g(x) there is no input that exists that will give us an output of _____.
Both functions can have inputs of _____ and give us two different outputs at that value for x.
Another feature the graphs allow us to see is that f(x) and g(x) have _____ points of intersection.
This means that there are two different ________ ________ that give us the _______ output
value from these functions.
Using technology lets us find an ______________ of those two points of intersection.
These points are at about: (-2.63, 1.05) and (1.71, 7.57).
Below x = -2.63 we can see on the graph that the output from g(x) is _______ than that from f(x).
Above x = -2.63, f(x) is _________ than g(x) until we get to about x = 1.71.
Above x = 1.71, the outputs from g(x) begin to ___________ very quickly and g(x) is once again
____________ than f(x).
Skill-based Task
1.
Which has a greater slope?
 f(x) = 3x – 5
 A function representing the
number of bottle caps in a
shoebox where 5 are added each
time.
Problem Task
1.
Create a graphic organizer to highlight
your understanding of functions and their
properties by comparing two functions
using at least two different
representations.