Quadratic function

From Wikipedia, the free encyclopedia

(Redirected from Quadratic curve)
Jump to: navigation, search
Merge arrows
 It has been suggested that this article or section be merged with Quadratic equation. (Discuss)
f(x) = x^2 - x - 2\,\!

A quadratic function, in mathematics, is a polynomial function of the form f(x)=ax^2+bx+c \,\!, where a \ne 0 \,\!.
It takes its name from the Latin quadratus for square, because quadratic functions arise in the calculation of the area of a square. A quadratic function is also referred to as a degree 2 polynomial or a 2nd degree polynomial, because the highest exponent of x is 2. The graph of such a function is a parabola.

If the quadratic function is set equal to zero, then the result is a quadratic equation. The solutions to the equation are called the roots of the equation or the zeros of the function.

Contents

The prefix quadri- is used to indicate the number 4. Examples are quadrilateral and quadrant. However, because it is in the Latin word for square (since a square has 4 sides), and the area of a square with side length x is x2, the prefix is also sometimes used in words involving the number 2.

The two roots of the quadratic equation 0=ax^2+bx+c\,\!, where a \ne 0 \,\! are

x = \frac{-b \pm \sqrt{b^2 - 4 a c}}{2 a}.

This formula is called the quadratic formula. To see how the formula is derived, see quadratic equation.

  • Let \Delta = b^2-4ac \,
  • If \Delta > 0\,\!, then there are two distinct roots since \sqrt{\Delta} is a positive real number.
  • If \Delta = 0\,\!, then the two roots are equal, since \sqrt{\Delta} is zero.
  • If \Delta < 0\,\!, then the two roots are complex conjugates, since \sqrt{\Delta} is imaginary.

By letting r_1 = \frac{-b + \sqrt{b^2 - 4 a c}}{2 a} and r_2 = \frac{-b - \sqrt{b^2 - 4 a c}}{2 a} or vice versa, one can factor a x^2 + b x + c \,\! as a(x - r_1)(x - r_2)\,\!.

A quadratic function can be expressed in three formats:

  • f(x) = a x^2 + b x + c \,\! is called the general form or polynomial form,
  • f(x) = a(x - r_1)(x - r_2)\,\! is called the factored form, where r1 and r2 are the roots of the quadratic equation, and
  • f(x) = a(x - h)^2 + k \,\! is called the standard form or vertex form.

To convert the general form to factored form, one needs only the quadratic formula to determine the two roots r1 and r2. To convert the general form to standard form, one needs a process called completing the square. To convert the factored form (or standard form) to general form, one needs to multiply, expand and/or distribute the factors.

Regardless of the format, the graph of a quadratic function is a parabola (as shown above).

  • If a > 0 \,\!, the parabola opens upward.
  • If a < 0 \,\!, the parabola opens downward.

The coefficient a controls the speed of increase (or decrease) of the quadratic function from the vertex, bigger positive a makes the function increase faster and the graph appear more closed.

The coefficients b and a together control the axis of symmetry of the parabola (also the x-coordinate of the vertex).

The coefficient b alone is the declivity of the parabola as it crosses the y-axis.

The coefficient c controls the height of the parabola, more specifically, it is the point were the parabola crosses the y-axis.

The x-intercepts of the graph are the same as the roots of the quadratic function (see above).

The vertex of a parabola is the place where it turns, hence, it's also called the turning point. If the quadratic function is in standard form, the vertex is (h, k)\,\!. By the method of completing the square, one can turn the general form: f(x) = a x^2 + b x + c \,\! to

f(x) = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2-4ac}{4 a} ,

so the vertex of the parabola in the general form will be

\left(-\frac{b}{2a}, -\frac{\Delta}{4 a}\right).

If the quadratic function is in factored form f(x) = a(x - r_1)(x - r_2) \,\!

the average of the two roots, i.e.,

\frac{r_1 + r_2}{2} \,\!

is the x-coordinate of the vertex, and hence the vertex is

\left(\frac{r_1 + r_2}{2}, f(\frac{r_1 + r_2}{2})\right).\!

The vertex is also the maximum point if a < 0 \,\! or the minimum point if a > 0 \,\!.

The vertical line

x=h=-\frac{b}{2a}

that passes through the vertex is also the axis of symmetry of the parabola.

  • Maximum and minimum points
The maximum or minimum of the function is always obtained at the vertex, the following method is another derivation of the same fact using calculus, the advantage of this method is that it works for more general functions.
Taking f(x) = ax^2 + bx + c \,\! as sample quadratic equation, to find its maximum or minimum points (which depends on a \,\!, if a > 0 \,\!, it has a minimum point, if a < 0\,\!, it has a maximum point) we have to first, take its derivative:
f(x)=ax^2+bx+c \Leftrightarrow \,\!f'(x)=2ax+b \,\!
Then, we find the roots of f'(x)\,\!:
2ax+b=0 \Rightarrow \,\! 2ax=-b \Rightarrow\,\! x=-\frac{b}{2a}
So, -\frac{b} {2a} is the x\,\! value of f(x)\,\!. Now, to find the y\,\! value, we substitute x = -\frac{b} {2a} on f(x)\,\!:
y=a \left (-\frac{b}{2a} \right)^2+b \left (-\frac{b}{2a} \right)+c\Rightarrow y= \frac{ab^2}{4a^2} - \frac{b^2}{2a} + c \Rightarrow y= \frac{b^2}{4a} - \frac{b^2}{2a} + c \Rightarrow
y= \frac{b^2 - 2b^2 + 4ac}{4a} \Rightarrow y= \frac{-b^2+4ac}{4a} \Rightarrow y= -\frac{(b^2-4ac)}{4a} \Rightarrow y= -\frac{\Delta}{4a}
Thus, the maximum or minimum point coordinates are:
\left (-\frac {b}{2a}, -\frac {\Delta}{4a} \right).

The square root of a quadratic function gives rise either to an ellipse or to a hyperbola.If a>0\,\! then the equation y = \pm \sqrt{a x^2 + b x + c} describes a hyperbola. The axis of the hyperbola is determined by the ordinate of the minimum point of the corresponding parabola y_p = a x^2 + b x + c \,\!
If the ordinate is negative, then the hyperbola's axis is horizontal. If the ordinate is positive, then the hyperbola's axis is vertical.
If a<0\,\! then the equation y = \pm \sqrt{a x^2 + b x + c} describes either an ellipse or nothing at all. If the ordinate of the maximum point of the corresponding parabola y_p = a x^2 + b x + c \,\! is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an empty locus of points.

A bivariate quadratic function is a second-degree polynomial of the form

f(x,y) = A x^2 + B y^2 + C x + D y + E x y + F \,\!

Such a function describes a quadratic surface. Setting f(x,y)\,\! equal to zero describes the intersection of the surface with the plane z=0\,\!, which is a locus of points equivalent to a conic section.

If 4AB-E^2 <0 \, the function has no maximum or minimum, its graph forms an hyperbolic paraboloid.

If 4AB-E^2 >0 \, the function has a minimum if A>0, and a maximum if A<0, its graph forms an elliptic paraboloid.

The minimum or maximum of a bivariate quadratic function is obtained at (x_m, y_m) \, where:

x_m = -\frac{2BC-DE}{4AB-E^2}
y_m = -\frac{2AD-CE}{4AB-E^2}

If 4AB- E^2 =0 \, and DE-2CB=2AD-CE \ne 0 \, the function has no maximum or minimum, its graph forms a parabolic cylinder.

If 4AB- E^2 =0 \, and DE-2CB=2AD-CE =0 \, the function achieves the maximum/minimum at a line. Similarly, a minimum if A>0 and a maximum if A<0, its graph forms a parabolic cylinder.

Retrieved from "http://en.wikipedia.org/wiki/Quadratic_function"
Advanced Search
Included Web Search Engines


Safe Search

close

Top Matching Results

Occasionally Search.com will highlight specialized results that are based on the context of your query. Examples of specialized results include specific links to news, images, or video.

Top Matching Results may highlight information from other Search.com pages, content from the CNET Network of sites, or third party content. The listings are based purely on relevance. Search.com does not receive payment for listings in this section but our partners that provide this data may get paid for listing these products.

Sponsored Links

This section contains paid listings which have been purchased by companies that want to have their sites appear for specific search terms and related content. These listings are administered, sorted and maintained by a third party and are not endorsed by Search.com.

Search Results

Search.com sends your search query to several search engines at one time and integrates the results into one list which has been sorted by relevance using Search.com's proprietary algorithm. You can customize the list of search engines included in your metasearch from the preferences.

The search engines that are used in your metasearch may allow companies to pay to have their Web sites included within the results. To view the Paid Inclusion policy for a specific search engine, please visit their Web site. Search.com does not accept payment or share revenue with any search engine partner for listings in this section.