Arent quadratic functions supposed to be symmetrical?

In summary, the conversation discusses the function f(x) and its derivative f'(x), as well as the point (0, f(0)). It also mentions the concept of symmetry in parabolas and how to find the minimum point using the completed square method.
  • #1
Jayden1
19
0
As we were talking about earlier.

f(x) = x^2 + 3x + 6
f ' (x) = 2x + 3

P1 = (0, f(0))

f ' (0) = 3.

This is not symetrical...

I hope you guys enjoy my question bombarding. As there are a lot of maths concepts that make me want to cry. It's funny, half the people in my degree fail their programming units and easily pass maths. However I easily pass my programming units and fail at maths. Sigh..
 
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  • #2
They are symmetrical about the line perpendicular to the directrix, and going through the focus. To find that, I would complete the square. In your case, you get

$$f(x)=x^{2}+3x+6=x^{2}+3x+\frac{9}{4}-\frac{9}{4}+6=\left(x+\frac{3}{2}\right)^{2}+\frac{15}{4}.$$

This is a parabola opening up - its minimum occurs at $x=-3/2$. Plug in that $x$-value to obtain the $f(-3/2)=15/4$. So the minimum occurs at $(-3/2,15/4)$. The parabola is symmetric about the line $x=-3/2$. See here.
 

FAQ: Arent quadratic functions supposed to be symmetrical?

Why are quadratic functions supposed to be symmetrical?

Quadratic functions are symmetrical because they can be represented by a parabola, which is a symmetrical shape. This means that the function has a line of symmetry or an axis of symmetry that divides the parabola into two equal halves.

How can I determine if a quadratic function is symmetrical?

A quadratic function is symmetrical if the leading coefficient (the coefficient of the x^2 term) is positive and the function has a quadratic term (x^2). This means that the function will have a parabola that opens either upwards or downwards, with a line of symmetry at the vertex.

Can a quadratic function still be symmetrical if the leading coefficient is negative?

No, a quadratic function with a negative leading coefficient will not be symmetrical. This is because the parabola will open downwards, and the axis of symmetry will not divide the parabola into two equal halves. However, it will still have one axis of symmetry at the vertex.

Are there any real-life applications of symmetrical quadratic functions?

Yes, symmetrical quadratic functions are commonly used in physics and engineering to model the trajectory of projectiles, such as a ball thrown into the air. They are also used in economics to model the cost and revenue functions of businesses.

What happens if a quadratic function is not symmetrical?

If a quadratic function is not symmetrical, it means that it does not have an axis of symmetry and the parabola is not evenly balanced. This can make it more difficult to analyze and solve the function, as the shape of the parabola will be distorted and not follow the usual patterns of symmetrical quadratic functions.

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