Graph Parabolas: Quadratic Equations Explained

In summary, to graph parabolas/quadratic equations, one can either use a calculator or plot points. The equation can be put into the form Y = a(x-h)^2 + k or X = a(y-h)^2 + h, where the vertex is located at (h, k), the focus is at (h, k+p), and the directrix line is y = k-p. The direction of the parabola opening can be determined by the value of a. To plot the graph, one can use a table of values or find the x and y intercepts. For more complex equations, a graphics calculator may be helpful.
  • #1
Imparcticle
573
4
How do I graph parabolas/quadratic equations?
 
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  • #2
Either use a calculator, or just plot points.

cookiemonster
 
  • #3
Ick, your making me remember how to graph stuff without calc?

Ill start off by assuming you mean a straight forward parabola without a rotation or anything.


First you need to put the parobala in the form:

Y = a(x – h)^2 + k
Or
X = a(y – h)^2 + h

Where a=1/4p

The position of your vertex (the extreme point) is (h,k)

The focus is (h, k + p) and the directrix line is y = k – p

The other 2nd degree quadratics have similar geometric formulas. Maybe you can show us what you need to graph…
 
Last edited:
  • #4
Oops forgot to tell you how to find which way it opens.

For
X = a(y – h)^2 + h
If a < 0 then it opens to the left
If a > 0 then it opens to the right

For
Y = a(x – h)^2 + k
If a < 0 then it opens down
If a > 0 then it opens up
 
  • #5
cookiemonster said:
Either use a calculator, or just plot points.

cookiemonster

I think it will be easier to just plug in points and go from there.
 
  • #6
Chrono said:
I think it will be easier to just plug in points and go from there.
to make a parabola graph you only need three points, if less then it's not kosher.
 
  • #7
Aside from the simple y=(x-a)^2+b formulas, there's also conics equations for parabolas:
4py=x^2, where p is the distance to your focus or directrix. (the focus is an arbirtrary point, and the directrix is a line whos equation is y=-d, where d is the distance from y=0 to focus).
As far as graphing, either get a simple table of values, or get three points: two X intercepts and one Y intercept. To get the X intercepts, simply substitute 0 into y so that your equation looks like 0 = (x-a)^2 + b, which shouldn't be too hard to solve. For the Y intercept, do the same, but substitute x as 0, so that you end up with y=(0-a)^2+b , which should be pretty easy to get as well.
after you're done that, plot your points, and draw a curve through them.
 
  • #8
how about graphing something like y=3x+5b+3?
 
  • #9
:eek: You have got more variables in there than I am am comfortable to deal with!
 
  • #10
did u mean y = 3x^2 + 5x + 3?

if u didnt:
thats just a simple linear equation (im assuming that b is a constant - if its a variable u can't graph it) that has a y-intercept of 5b + 3 and a slope of 3..
 
  • #11
loop quantum gravity said:
to make a parabola graph you only need three points, if less then it's not kosher.

You're right, of course. For some reason I was thinking of a line when I said what I did.
 
  • #12
Parth Dave said:
did u mean y = 3x^2 + 5x + 3?

Yes. (my bad) :rolleyes:
 
  • #13
for that kind i just use a graphics calculator, i think if you want to look at that and draw it you have to play with the equation a bit till its easier. i 4got the format i used last year for this type of thing.
 
  • #14
If b is a third variable (call it z, for familiarity), then I believe you are looking at the equation of a plane.
 
1.

What is a parabola and how does it relate to quadratic equations?

A parabola is a U-shaped curve that is formed when graphing a quadratic equation. It is the visual representation of the relationship between the x and y variables in a quadratic equation. The equation for a parabola is y = ax^2 + bx + c, where a, b, and c are constants.

2.

What does the shape of a parabola tell us about the roots of a quadratic equation?

The shape of a parabola can tell us whether the quadratic equation has two distinct real roots, one real root, or no real roots. If the parabola opens upwards, it indicates that the equation has a minimum value and two distinct real roots. If the parabola opens downwards, it indicates that the equation has a maximum value and no real roots. If the parabola is a straight line, it indicates that the equation has one real root.

3.

How do we find the vertex of a parabola?

The vertex of a parabola is the highest or lowest point on the curve, depending on whether it opens upwards or downwards. To find the vertex, we can use the formula x = -b/2a, where a and b are the coefficients of the quadratic equation. Once we find the value of x, we can plug it back into the equation to find the y-coordinate of the vertex.

4.

What is the significance of the discriminant in quadratic equations?

The discriminant, b^2 - 4ac, is a term that appears in the quadratic formula. It tells us whether the quadratic equation has two distinct real roots, one real root, or no real roots. If the discriminant is greater than 0, the equation has two distinct real roots. If it is equal to 0, the equation has one real root. If it is less than 0, the equation has no real roots.

5.

How can we use the graph of a parabola to solve quadratic equations?

We can use the graph of a parabola to solve quadratic equations by identifying the x-intercepts, which are the points where the parabola intersects the x-axis. These points represent the solutions to the equation. We can also use the vertex of the parabola to find the minimum or maximum value of the equation, which can be useful in real-world applications.

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