Function Graphs - Exercises | IB Math AAHL | Perplex

Function Graphs

Graphing functions with a calculator, characteristics of a function, even and odd functions, x and y intercepts, horizontal and vertical asymptotes, maxima and minima

Want a deeper conceptual understanding? Try our interactive lesson!

Exercises

No exercises available for this concept.

Key Skills

Graph of a function

SL Core 2.3

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.

Vertical line test

SL Core 2.2

As mentioned earlier, a function is a special type of relation where each x value has only one possible y-value.

Graphically, this means functions pass the vertical line test, which means that no vertical line intersects the graph of a function twice:

Finding function values from graph

SL Core 2.2

We can find the value f(a) of a function by looking at the y-value of the graph where x=a:

Graphing with technology

SL Core 2.3

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.

Intersections with GDC

SL Core 2.4

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.

x-intercepts

SL Core 2.4

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.

y-intercepts

SL Core 2.4

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.

Calculator: finding value of a function

SL Core 2.3

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) and GRAPH it. Then, 2nd > CALC > VALUE and input a when prompted X=. This returns 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).

Horizontal asymptotes

SL Core 2.3

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.

Vertical asymptotes

SL Core 2.3

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

f(x)=h(x)g(x)

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.

Maxima and Minima

SL Core 2.4

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.

Finding maxima & minima with technology

SL Core 2.4

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.

Even functions

AHL 2.14

An even function is one for which

f(−x)=f(x) for all x∈R

Graphically, this means the function is symmetric in the y-axis:

Odd functions

AHL 2.14

An odd function is one for which

f(−x)=−f(x) for all x∈R

Graphically, this means the function has pointwise symmetry with the origin:

Graph of a function

SL Core 2.3

Vertical line test

SL Core 2.2

As mentioned earlier, a function is a special type of relation where each x value has only one possible y-value.

Graphically, this means functions pass the vertical line test, which means that no vertical line intersects the graph of a function twice:

Finding function values from graph

SL Core 2.2

We can find the value f(a) of a function by looking at the y-value of the graph where x=a:

Graphing with technology

SL Core 2.3

Intersections with GDC

SL Core 2.4

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.

x-intercepts

SL Core 2.4

y-intercepts

SL Core 2.4

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.

Calculator: finding value of a function

SL Core 2.3

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) and GRAPH it. Then, 2nd > CALC > VALUE and input a when prompted X=. This returns f(a).

Horizontal asymptotes

SL Core 2.3

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.

Vertical asymptotes

SL Core 2.3

Vertical asymptotes have the equation x=a for some a∈R.

Vertical asymptotes occur for functions of the form

f(x)=h(x)g(x)

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.

Maxima and Minima

SL Core 2.4

The global maxima and minima are the overall maximum or minimum points on the curve.

Finding maxima & minima with technology

SL Core 2.4

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.

Even functions

AHL 2.14

An even function is one for which

f(−x)=f(x) for all x∈R

Graphically, this means the function is symmetric in the y-axis:

Odd functions

AHL 2.14

An odd function is one for which

f(−x)=−f(x) for all x∈R

Graphically, this means the function has pointwise symmetry with the origin: