How many points determine a parabola

We can rationalize it with the transform. The detailed discussion of the number of real roots seems to be an endeavor. Four points determine two unique parabolas as mentioned by ccorn anyway you wish to place them, subject to convexity and other conditions to avoid degeneracy also as stated by him. There is a doubly infinite set, a new rough sketch indicates both. So from the above if you choose one rigid parabolic arc among them, then there is a unique way to fit it back after removing from the 3 given points to re-assemble it.

Along the parabola any motion leaves it unchanged proving that the drawn curve is indeed a parabola.. Shown here are three for each set but there are infinitely many for each. A parabola is a conic with a double point at infinity. For each point on the line at infinity other than the three points corresponding to the three lines determined by the given three points, there is a unique parabola through the given three points not on a line.

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Learn more. How many parabolas can be formed from 3 points? If we allow rotation Ask Question. Asked 4 years, 5 months ago. Active 4 years, 5 months ago. Viewed 6k times. A quick sketch of what I mean sorry for the poor drawing accuracy Here we have 3 parabolas being formed by the same 3 distinct points. Stephen Stephen 3, 1 1 gold badge 13 13 silver badges 29 29 bronze badges.

In a comment, I provide an equation for these parabolas, but I never did get around to showing the derivation.

Almost worth reposting here! Show 2 more comments. Active Oldest Votes. Harambe Harambe 7, 2 2 gold badges 26 26 silver badges 48 48 bronze badges. Of course that doesn't really change your argument, as there are only three such exceptions. Since rotations map straight lines to straight lines, collinearity is preserved by them. All but strictly necessary cookies are currently disabled for this browser. Turn on JavaScript to exercise your cookie preferences for all non-essential cookies.

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Learn more about this course. Step 1: Determine the y -intercept. The y -intercept is 0, 3. Step 2: Determine the x -intercepts. Step 3: Determine the vertex. Step 4: Determine extra points so that we have at least five points to plot. In this example, one other point will suffice. Step 5: Plot the points and sketch the graph. To recap, the points that we have found are. The parabola opens downward. In general, use the leading coefficient to determine whether the parabola opens upward or downward.

If the leading coefficient is negative, as in the previous example, then the parabola opens downward. If the leading coefficient is positive, then the parabola opens upward.

However, not all parabolas have x intercepts. Solution: Because the leading coefficient 2 is positive, note that the parabola opens upward. Use the discriminant to determine the number and type of solutions. Since the discriminant is negative, we conclude that there are no real solutions.

Because there are no real solutions, there are no x -intercepts. Next, we determine the x -value of the vertex. So far, we have only two points. Plot the points and sketch the graph. The x -value of the vertex can be calculated as follows:. Given that the x -value of the vertex is 3, substitute into the original equation to find the corresponding y -value.

Therefore, the vertex is 3, 0 , which happens to be the same point as the x -intercept. Choose x -values 1, 5, and 6. Here we obtain two real solutions for x , and thus there are two x -intercepts:.

Approximate values using a calculator:. Use the approximate answers to place the ordered pair on the graph. However, we will present the exact x -intercepts on the graph. Next, find the vertex. Given that the x -value of the vertex is 1, substitute into the original equation to find the corresponding y -value.

We need one more point. Try this! To find these important values given a quadratic function, we use the vertex. If the leading coefficient a is positive, then the parabola opens upward and there will be a minimum y -value. If the leading coefficient a is negative, then the parabola opens downward and there will be a maximum y -value. To find it, we first find the x -value of the vertex. The x -value of the vertex is 3. Substitute this value into the original equation to find the corresponding y -value.

The vertex is 3, 1. Begin by finding the x -value of the vertex. A parabola, opening upward or downward as opposed to sideways , defines a function and extends indefinitely to the right and left as indicated by the arrows.

Therefore, the domain the set of x -values consists of all real numbers.