Graphing functions with a calculator, characteristics of a function, even and odd functions, x and y intercepts, horizontal and vertical asymptotes, maxima and minima
A graph is a powerful way to visualize the behavior of a function. To see how, let's start with a super basic example:
From the diagram above we see that
f(6)=3
f(8)=0
f(−9)=−1
If we call the input of the function x, and name the output y=f(x), then each input-output pair becomes a point:
We can then plot these points on the cartesian plane:
In summary, the graph of a function shows all the points (input,output), with the x-axis representing the inputs and the y-axis representing the outputs.
The graph is a visual representation of a function made by placing all the points (x,f(x)) on a coordinate plane. Since most functions have infinitely many possible inputs x, the graph often looks like a line.
We can find the value f(a) of a function by looking at the y-value of the graph where x=a:
To test if a graph represents a function, we use the vertical line test. If any vertical line intersects the graph in two locations, the graph cannot be of a function.
When the graph of a function is not known, a calculator can be used to plot its curve. Under the hood, the calculator is simply plotting points at regular x intervals, and connecting them with straight lines. For a small enough interval, this approximation is basically unnoticeable.
Calculators are able to provide precise information on coordinates of an intersection of two functions.
Using a TI-84, 2nd > CALC > INTERSECT,
or using a Casio, SHIFT > G-SOLV > INTSECT
will find an intersection between two functions.
The x-intercepts of a function are the values of x where the curve intersects the x-axis. Since the x-axis has the equation y=0, this means the function is equal to zero. For this reason, x-intercepts are also often called the "zeros" of the function.
The y-intercept of a function is the values of y where the curve intersects the y-axis. Since the y-axis has the equation x=0, we can plug 0 into the function definition to find the intercept.
On a graphing calculator, you can quickly calculate f(a) for any function f(x) and any a in its domain.
On a TI-84, under Y= set Y1=f(x), then on the home screen type vars > Y-VARS > FUNCTION > Y1, enter Y1(a), and hit enter to see the value of f(a).
On a Casio, from the calculate app, press FUNCTION, choose Define f(x) and input your equation. Then, press FUNCTION again and now press f(x) , and type in the value of a and a close parenthesis. This returns f(a).
A horizontal asymptote is a horizontal line that a graph approaches but never actually touches or crosses. It's seen where a function approaches a certain constant value as x gets very large (positive or negative).
Horizontal asymptotes have the equation y=a for some a∈R.
To find a horizontal asymptote with a calculator, plot the curve and use trace to inspect the value of y for large (positive or negative) x.
A vertical asymptote is a vertical line that a graph approaches but never actually touches or crosses. It's usually seen where a function "blows up," meaning the function’s values become infinitely large or small as you get closer to certain values of x.
Vertical asymptotes have the equation x=a for some a∈R.
Vertical asymptotes occur for functions of the form
when h(x)=0 and g(x)=0, as division by zero is undefined, and division by a number close to zero gives a large number.
A curve has local maxima or minima when the curve changes y-direction. This means that a local maximum is greater than the points immediately to its left and right on the graph; likewise, a local minimum is greater than the points immediately to its left and right on the graph.
The global maxima and minima are the overall maximum or minimum points on the curve.
Graphing calculators can be used to find the coordinates of maxima and minima of a function. On a TI-84, click 2nd > CALC and then either 3:minimum or 4:maximum.
Before trying to find a minimum or maximum, check visually whether the function has the desired extrema.
Nice work completing Function Graphs, here's a quick recap of what we covered:
Exercises checked off
Graphing functions with a calculator, characteristics of a function, even and odd functions, x and y intercepts, horizontal and vertical asymptotes, maxima and minima
A graph is a powerful way to visualize the behavior of a function. To see how, let's start with a super basic example:
From the diagram above we see that
f(6)=3
f(8)=0
f(−9)=−1
If we call the input of the function x, and name the output y=f(x), then each input-output pair becomes a point:
We can then plot these points on the cartesian plane:
In summary, the graph of a function shows all the points (input,output), with the x-axis representing the inputs and the y-axis representing the outputs.
The graph is a visual representation of a function made by placing all the points (x,f(x)) on a coordinate plane. Since most functions have infinitely many possible inputs x, the graph often looks like a line.
We can find the value f(a) of a function by looking at the y-value of the graph where x=a:
To test if a graph represents a function, we use the vertical line test. If any vertical line intersects the graph in two locations, the graph cannot be of a function.
When the graph of a function is not known, a calculator can be used to plot its curve. Under the hood, the calculator is simply plotting points at regular x intervals, and connecting them with straight lines. For a small enough interval, this approximation is basically unnoticeable.
Calculators are able to provide precise information on coordinates of an intersection of two functions.
Using a TI-84, 2nd > CALC > INTERSECT,
or using a Casio, SHIFT > G-SOLV > INTSECT
will find an intersection between two functions.
The x-intercepts of a function are the values of x where the curve intersects the x-axis. Since the x-axis has the equation y=0, this means the function is equal to zero. For this reason, x-intercepts are also often called the "zeros" of the function.
The y-intercept of a function is the values of y where the curve intersects the y-axis. Since the y-axis has the equation x=0, we can plug 0 into the function definition to find the intercept.
On a graphing calculator, you can quickly calculate f(a) for any function f(x) and any a in its domain.
On a TI-84, under Y= set Y1=f(x), then on the home screen type vars > Y-VARS > FUNCTION > Y1, enter Y1(a), and hit enter to see the value of f(a).
On a Casio, from the calculate app, press FUNCTION, choose Define f(x) and input your equation. Then, press FUNCTION again and now press f(x) , and type in the value of a and a close parenthesis. This returns f(a).
A horizontal asymptote is a horizontal line that a graph approaches but never actually touches or crosses. It's seen where a function approaches a certain constant value as x gets very large (positive or negative).
Horizontal asymptotes have the equation y=a for some a∈R.
To find a horizontal asymptote with a calculator, plot the curve and use trace to inspect the value of y for large (positive or negative) x.
A vertical asymptote is a vertical line that a graph approaches but never actually touches or crosses. It's usually seen where a function "blows up," meaning the function’s values become infinitely large or small as you get closer to certain values of x.
Vertical asymptotes have the equation x=a for some a∈R.
Vertical asymptotes occur for functions of the form
when h(x)=0 and g(x)=0, as division by zero is undefined, and division by a number close to zero gives a large number.
A curve has local maxima or minima when the curve changes y-direction. This means that a local maximum is greater than the points immediately to its left and right on the graph; likewise, a local minimum is greater than the points immediately to its left and right on the graph.
The global maxima and minima are the overall maximum or minimum points on the curve.
Graphing calculators can be used to find the coordinates of maxima and minima of a function. On a TI-84, click 2nd > CALC and then either 3:minimum or 4:maximum.
Before trying to find a minimum or maximum, check visually whether the function has the desired extrema.
Nice work completing Function Graphs, here's a quick recap of what we covered:
Exercises checked off