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### Course: Algebra (all content)>Unit 7

Lesson 24: Determining the range of a function (Algebra 2 level)

# Domain and range of quadratic functions

Sal finds the domain and the range of f(x)=3x^2+6x-2. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• how can we know where is the vertex?
• Sal mentions how to find the vertex at . Its a part of the Quadratic Formula: -b/2a
• Should I take the Quadratics course before Functions?
• box 0f rox, great question! It's really your choice. I think I would recommend going through functions first, if you can, just because functions are really fundamental to all types of equations, including quadratic equations. Also, Sal has the funcitons playlist before quadratics, which means that he probably assumes you have understood the idea behind functions when he explains quadratics.
Hope that helps!
• why do quadratics have this behavior? i mean why is their graph always parabolic?
• Well, that's what the squaring function does. It produces a parabolic graph (not a hyperbola). So any quadratic (which includes a squared term) has the same sort of graph.
• At Sal says there are Imaginary Numbers, can someone please give me an example of one?
• The imaginary constant, i, is the principal square root of -1 ( i = +(-1)^1/2 ). Any multiple of i is a imaginary number.
Examples of imaginary numbers: i, 19i, 27i
The real numbers are the counting, negative, 0, rational, and irrational numbers.
Example of real numbers : 3.1415926535... (pi), 2.718281828459045...(e), 1
When you combine real numbers with imaginary numbers, you get complex numbers.
Examples of complex numbers: 17 + 5i, 29 + 7i, 81 + 0i ( real numbers )
Hope this helps!
• Will there ever be any not real numbers on this or no?
• Good question!

No, because it must be on an number line. It is impossible to have a domain not on an number line.

Technically, anything is an "real number". Intervals, integers, fractions, whole numbers, negatives, irrational, rational. I could make a Venn Diagram off a hundred circles depicting this, but in the end, surrounding all those circles, is one infinite circle known as "Real Numbers"

Hope you enjoyed!
• How does the fact that plugging in -b/2a will give you the vertex logically make sense? I'm aware that it's a piece of the quadratic formula, but why that particular part?
• That the x-coordinate of a vertex of a parabola is always -b/2a is derived by using calculus. The actual computations I won't cover here, but let us just say that the vertex of a parabola is the only point of the parabola where the ever-changing slope equals zero.

You have to use calculus to get this function, but let us just say that for the equation y = ax² + bx + c, the slope is the function m = 2ax + b
If you set that equal to 0 and solve for x, then you have to point in the parabola with zero slope. That will be:
2ax + b = 0
2a x = - b
x = - b / 2a

Oh, and the y-coordinate of the vertex of a parabola is always at c - (b² / 4a)
• What does he mean by 'completing the square' at
• How would you find the domain & range of a quadratic in vertex form ?
• For every polynomial function (such as quadratic functions for example), the domain is all real numbers.
If f(x) = a(x-h)² + k , then
if the parabola is opening upwards, i.e. a > 0 , the range is y ≥ k ;
if the parabola is opening downwards, i.e. a < 0 , the range is y ≤ k .
• how do i do an equation that looks like this:
determine the domain and range of Y= -3x^ -9?