Functions and Graphs 2 Flashcards
What is ALWAYS used to write where a graph is increasing, decreasing, or remains constant.
Open intervals on the x-axis (because behavior at endpoints is excluded).
From what to what direction should evaluation of a graph happen when determining the increasing, decreasing, or constant states of a graph?
From left to right.
What’s a relative maximum in a graph?
A relative maximum is the greatest y value in a local(small) area of a graph. It is called relative, or local, because it doesn’t relate to the whole graph but for a small portion of it instead.
What is reffered to as a relative “peak” in graphs?
The relative maximum of that specific area.
What’s a relative minimum?
A point on a graph where the graph goes from decreasing to increasing. Aka the lowest point.
What’s used as an answer when asked for the location of a relative maximum/minimum?
The x value of that specific point.
What’s used as an answer when asked for the relative maximum of a specific point?
The y value of that point.
What does the word “symmetria” mean?
The same measure.
Find the relative maximum and its location for the function f(x) = x^3-3x.
Relative maximum location -1, the value of the relative maximum = 2.
What’s the definition for a symmetric graph with respect to the y-axis?
A graph is symmetric with respect to the y-axis if for every (x,y) point on the graph, the point (-x,y) is also on the graph.
What’s the definition for a symmetric graph with respect to the x-axis?
A graph is symmetric with respect to the x-axis if for every point (x,y) on the graph, the point (x,-y) is also on the graph.
What’s the definition for a symmetric graph with respect to the origin?
A graph is symmetric with respect to the origin if for every (x,y) point, the point (-x,-y) is also on the graph.
With respect to what is the graph of the function f(x) symmetric if f(-x) = f(x)?
To the y-axis.
Using the equation of a function, how do we test if a function is symmetric with respect to the y-axis? Use f(x) = x^2 for your explanation.
Replace x with -x in the equation, if the new equation is equivalent, then the function’s graph is symmetrical in respect to the y-axis.
y = x^2,
y = (-x)^2 = x^2,
(y=x^2) = (y=(-x^2)), the function’s graph is symmetrical to the y-axis.
What type of symmetric graph is not a function?
Graph symmetric with respect to the x-axis.
Using the equation of a graph, how do we test if it is symmetric with respect to the origin?
Use this equation for your example: y^2 = x^2 + 2
We substitute x with -x and y with -y, if we get an equivalent equation, then the graph is symmetric with respect to the origin.
How is a function called if its graph is symmetric with respect to the y-axis?
An even function.
How is an even function defined? Give an example with a function that satisfies the definition.
The function f is an even function if f(-x) = f(x).
Example: f(x) = x^2, f(-x) = (-x)^2 = x^2, f(x) = f(-x).
How is a function called if its graph is symmetric with respect to the origin.
An odd function.
What’s the definition for an odd function? Give an example with a function that satisfies the definition.
The function f is odd if f(-x) = -f(x).
Example: f(x) = x^3 + x,
f(-x) = -x^3 - x,
-f(x) = -(x^3 + x), f(x) = -x^3 - x,
-f(x) = f(-x).
Instead of using the definition for an odd function, what’s a quicker way to determine if a function is odd?
By looking at the function’s equation’s terms. If each term’s exponent is an odd number, then this function is odd.
Instead of using the definition for an even function, what’s a swifter way to determine if a function is even?
By looking at the function’s equation’s terms. If each term’s exponent is an even number, then this function is even.
What’s a piecewise function?
A function that is defined by two or more equations over a specified domain.
What’s a “greatest integer function”?
A function that gives the greatest integer which is less than or equal to x.
