In order to maintain the methodology of my courses, I now highlight the following aspects:

Definition:

Simply put, an equalizer is an effect that enables us to adjust the volume in the frequency range. Unlike a fader, as this raises or lowers all frequencies equally.
The working range of any EQ is limited by our human hearing, which ranges from approx. 20 Hz to 20,000 Hz. It is important to note that you can only amplify with a frequency band what is present in the signal.

Various forms of EQs

I make a rough distinction between the following working methods

Clean EQs that do what they indicate and generate few harmonics (usually stock EQs or parametric equalizers)
Vintage EQs that additionally integrate sound color through saturation (such as the SSL Channel Strip or the Pultec EQ)
dynamic EQ's, which allow you to influence the dynamics of each band
creative EQ's and filters, for sound design (plug-in "Volcano3")
Resonance suppressors or AI EQs (Sooth2)

Intentions

Removing unwanted frequencies: I would use a technical EQ to remove unwanted frequencies such as rumble, hum or background noise, as it reliably does exactly what I tell it to do. High and low cuts or shelves are common EQ techniques that can be used for this purpose. However, it is important not to cut every signal for no reason, as this can lead to a thin, characterless mix.

Shaping sound character: One of the main functions of an EQ is to balance the tonal characteristics of an audio signal. By boosting or attenuating certain frequency ranges, you can make a sound brighter (by boosting the high frequencies), warmer (by boosting the low frequencies) or more neutral.

Frequency correction: EQs are used to correct or compensate for problems during recording or mixing. For example, if a microphone has recorded a voice with excessive sibilance (hard "s" and "sibilant" sounds), an EQ can be used to reduce these frequencies. The use of a dynamic EQ is particularly effective here in order to only react when the sibilants are present.

Improving clarity: You can use EQ to improve the clarity and definition of individual instruments or voices in a mix. While boosting in the 1-3 kHz range can optimize the intelligibility of vocals and the presence of instruments, the most important thing to understand is that you can achieve the same end result by lowering the lower mids. Many engineers swear by the idea that you should mainly cut with an EQ. However, I have discussed this in another blog post.

The eternal battle in the frequency range: In mixing, different instruments can occupy the same frequency range and compete for sound space. The EQ can help to find enough space for the individual instruments by cutting or attenuating frequencies where they meet. Here too, however, I would advise

Special effects: In addition to correction and equalization purposes, the EQ can also be used creatively to achieve special effects. For example, boosting and sweeping a narrow frequency band can create filter sweeps or resonance effects.

Room correction: In recording studios, EQs can be used in room correction systems to compensate for acoustic anomalies in the listening environment. I use Sonarworks for this.

Stereo imaging: During mastering, the mid-side EQ can be used to adjust the stereo image of a mix and improve the width and depth of the sound field.

It is important to use EQ wisely and with a specific goal/intention. Overuse of EQ can lead to unnatural or undesirable results. Experienced mixing engineers and producers have a keen sense of the subtle adjustments needed to achieve the desired sound. However, the professionals always follow one principle: as much EQ as necessary and as little processing as possible.