Function Analysis AP Precalculus AD2024

FUNCTION ANALYSIS AND GRAPH

How to determine the important elements of a function (analytically)

BEGIN

By profr. Edgar Orozco, 2022 (modified 2024)

Index

Analysis of functions

General Elements

6) Graph

3) Symmetry

4) Behavior near restrictions

7) Range

1) Domain

5) End behavior

Summary

2) x and y intercepts

INTRODUCTION

General elements

Analyzing a function means to determine all of its important elements in order to sketch a more accurate graph. The basic elements of a function are: 1) Domain2) x and y intercepts 3) Symmetry3) Behavior near the restriction 4) End behavior5) Graph6) Range

1) Restrictions and domain

Depending of the type of function, the restrictions (values of x that cannot be taken) change:

• Polynomial function: No restrictions
• Rational function: Denominator ≠ 0
• Radical (even power) function: Radicand ≥ 0
• Exponential function: No restrictions
• Logarithmic function: Argument > 0

2) x and y intercepts

y-intercept: Intersection point between the function and the y-axis. Found by letting 𝑥=0 and evaluating 𝑓(0).

x-intercepts: Intersection points between the function and the x-axis. Found by letting 𝑓(𝑥)=0 and isolating x.

+ examples

+ examples

3) Symmetry

Examples

4) Behavior near the restrictions

Once restrictions are identified (for this explanation, the restriction is 𝑥=𝑎):

Click on the buttons to check the following steps.

If 𝑥=𝑎 is a possible VA:

If 𝑎 is the possible x-coordinate of an empty dot

5) End behavior

Analyze BOTH the limits:

No HA, but...

HA: y = c

Asymptotic behavior

If there is no H.A., but the degree of the numerator is greater than the degree of the denominator, you may have asymptotic behavior.

• Use division of polynomials (dividing numerator by the denominator) to obtain:

where y = Q(x) (the quotient) is the asymptotic function.

EXAMPLE

6) GRAPH

To graph, use all the obtained elements:

• Plot the x and y intercepts.
• Draw the horizontal / slant / vertical asymptotes.
• Plot the empty dots.
• Start sketching the graph using the limits as indicators of the behavior of the graph.
• You may evaluate more values to make the graph as exact as possible.

7) Range

To determine the range, analyze the graph (from BOTTOM to TOP), taking into account:

• Empty dots
• Any horizontal asymptotes found in it.

Summary

Steps to analyze a function

Step 1

Step 2

Step 3

Step 4

Analyze the symmetry

Define x and y intercepts

Analyze the behavior near the restrictions

Analyze restrictions and find the domain

Summary (continuation)

Function analysis

Step 5

Step 6

WIN!

Step 7

Analyze the end behavior

Graph the function with all of its elements

Define the range

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