determining the key features of function graphs 10 february 2011
TRANSCRIPT
Determining the Key Features of Function
Graphs
10 February 2011
The Key Features of Function Graphs - Preview
Domain Range x-intercepts y-intercept End Behavior
Intervals of increasing, decreasing, and constant behavior Parent Equation Maxima and Minima
Domain Reminder: Domain is the set of all
possible input or x-values When we find the domain of the graph
we look at the x-axis of the graph
Determining Domain - Symbols Open Circle → Exclusive ( )
Closed Circle → Inclusive [ ]
Determining Domain1. Start at the origin2. Move along the x-axis until you find the
lowest possible x-value. This is your lower bound.
3. Return to the origin4. Move along the x-axis until you find your
highest possible x-value. This is your upper bound.
Examples
Domain:Domain:
Example
Domain:
Determining Domain - Infinity
Domain:
Examples
Domain: Domain:
Your Turn: In the purple Precalculus textbooks,
complete problems 3, 7, and find the domain of 9 and 10 on pg. 160
3. 7.
9. 10.
Range The set of all possible output or y-
values When we find the range of the graph
we look at the y-axis of the graph We also use open and closed circles for
the range
Determining Range Start at the origin Move along the y-axis until you find the
lowest possible y-value. This is your lower bound.
Return to the origin Move along the y-axis until you find your
highest possible y-value. This is your upper bound.
Examples
Range: Range:
Examples
Range: Range:
Your Turn: In the purple Precalculus textbooks,
complete problems 4, 8, and find the domain of 9 and 10 on pg. 160
4. 8.
9. 10.
X-Intercepts Where the graph crosses the x-axis Has many names:
x-intercept Roots Zeros
Examples
x-intercepts: x-intercepts:
Y-Intercepts Where the graph crosses the y-axis
y-intercepts: y-intercepts:
Seek and Solve!!!
Roller Coasters!!!
Fujiyama in Japan
Types of Behavior – Increasing As x increases, y also increases Direct Relationship
Types of Behavior – Constant As x increases, y stays the same
Types of Behavior – Decreasing As x increases, y decreases Inverse Relationship
Identifying Intervals of Behavior We use interval notation The interval measures x-values. The type
of behavior describes y-values.Increasing: [0, 4)
The y-values are increasing
when the x-values are between 0 inclusive and 4 exclusive
Identifying Intervals of Behavior Increasing:
Constant:
Decreasing:
x
1
1
y
Identifying Intervals of Behavior, cont. Increasing:
Constant:
Decreasing:-1-3
y
x
Don’t get distracted by the arrows! Even though both of the arrows point “up”, the graph isn’t increasing at both ends of the graph!
Your Turn: Complete problems 1 – 4 on The Key
Features of Function Graphs – Part II handout.
1.
2.
3.
4.
What do you think of when you hear the word parent?
Parent Function The most basic form of a type of function Determines the general shape of the
graph
Basic Types of Parent Functions1. Linear2. Absolute Value3. Greatest Integer4. Quadratic
5. Cubic6. Square Root7. Cube Root8. Reciprocal
Function Name: Linear Parent Function: f(x) = x
“Baby” Functions: f(x) = 3x f(x) = x + 6 f(x) = –4x – 2
y
x2
2
Greatest Integer Function f(x) = [[x]] f(x) = int(x) Rounding function
Always round down
“Baby” Functions Look and behave similarly to their parent
functions To get a “baby” functions, add, subtract,
multiply, and/or divide parent equations by (generally) constants f(x) = x2 f(x) = 5x2 – 14 f(x) = f(x) = f(x) = x3 f(x) = -2x3 + 4x2 – x + 2
x1
x24
Your Turn: Create your own “baby” functions in your
parent functions book.
Identifying Parent Functions From Equations Identify the most important operation
1. Special Operation (absolute value, greatest integer)
2. Division by x3. Highest Exponent (this includes square roots
and cube roots)
Examples1. f(x) = x3 + 4x – 3
2. f(x) = -2| x | + 11
3. ]]x[[)x(f 2
Identifying Parent Equations From Graphs Try to match graphs to the closest parent
function graph
Examples
Your Turn: Complete problems 5 – 12 on The Key
Features of Function Graphs handout
Maximum (Maxima) and Minimum (Minima) PointsPeaks (or hills) are your
maximum points
Valleys are your minimum points
Identifying Minimum and Maximum Points Write the answers as
points You can have any
combination of min and max points
Minimum:
Maximum:
Examples
Your Turn: Complete problems 1 – 6 on The Key
Features of Function Graphs – Part III handout.
Reminder: Find f(#) and Find f(x) = x
Find f(#) Find the value of f(x)
when x equals #. Solve for f(x) or y!
Find f(x) = # Find the value
of x when f(x) equals #.
Solve for x!
Evaluating Graphs of Functions – Find f(#)
1. Draw a (vertical) line at x = #
2. The intersection points are points where the graph = f(#)
f(1) = f(–2) =
Evaluating Graphs of Functions – Find f(x) = #
1. Draw a (horizontal) line at y = #
2. The intersection points are points where the graph is f(x) = #
f(x) = –2 f(x) = 2
Example
1. Find f(1)
2. Find f(–0.5)
3. Find f(x) = 0
4. Find f(x) = –5
Your Turn: Complete Parts A – D for problems 7 – 14
on The Key Features of Function Graphs – Part III handout.