អនុគមន៍ដឺក្រេទី២

គោលបំណងនៃមេរៀន

ក្រោយពីសិក្សាមេរៀននេះចប់ អ្នកសិក្សាទាំងគោលបំណងចម្បងនៃមេរៀននេះគឺ៖

  1. ចង់អោយស្គាល់អោយបានច្បាស់ពី រូបរាងស្តង់ដារបស់អនុគមន៍ដឺក្រេទី២ និងក្រាបតំណាងអោយអនុគមន៍នេះ
  2. ចង់អោយសិស្សអាចយល់បានពីទ
  3. ចង់អោយសិស្ស

សេចក្តីផ្តើម

អនុគមន៍ដឺក្រេទី២ មានរាងទូទៅគឺ៖ y = f(x) = ax2 + bx + c = 0   ដែល  a, b ​​​ និង c គឺជាមេគុណ និងគោរពលក្ខខណ្ឌ a ≠0។អនុគមន៍នេះ មានក្រាប ឬក្រាហ្វិកតាងអនុគមន៍ជារាងប៉ារាបូលដូចរូបខាងក្រោម។

ក្រាបអនុគមន៍ y=x2។គួរបញ្ជាក់ថា អនុគមន៍នេះមានមេគុណ a=1, b=0 និង c=0។

Quadratic 1

The function of the coefficient aa in the general equation is to make the parabola “wider” or “skinnier”, or to turn it upside down (if negative):

If the coefficient of x2x2 is positive, the parabola opens up; otherwise it opens down.

The Vertex

The vertex of a parabola is the point at the bottom of the ” UU ” shape (or the top, if the parabola opens downward).

The equation for a parabola can also be written in “vertex form”:

y=a(xh)2+ky=a(x−h)2+k

In this equation, the vertex of the parabola is the point (h,k)(h,k) .

You can see how this relates to the standard equation by multiplying it out:

y=a(xh)(xh)+ky=a(x−h)(x−h)+k

y=ax22ahx+ah2+ky=ax2−2ahx+ah2+k

The coefficient of xx here is 2ah−2ah . This means that in the standard form, y=ax2+bx+cy=ax2+bx+c , the expression

b2a−b2a

gives the xx -coordinate of the vertex.

Example:

Find the vertex of the parabola.

y=3x2+12x12y=3×2+12x−12

Here, a=3a=3 and b=12b=12 . So, the xx -coordinate of the vertex is:

122(3)=2−122(3)=−2

Substituting in the original equation to get the yy -coordinate, we get:

y=3(2)2+12(2)12y=3(−2)2+12(−2)−12

=24=−24

So, the vertex of the parabola is at (2,24)(−2,−24) .

The Axis of Symmetry

The axis of symmetry of a parabola is the vertical line through the vertex. For a parabola in standard form, y=ax2+bx+cy=ax2+bx+c , the axis of symmetry has the equation

x=b2ax=−b2a

Note that b2a−b2a is also the xx -coordinate of the vertex of the parabola.

Example:

Find the axis of symmetry.

y=2x2+x1y=2×2+x−1

Here, a=2andb=1a=2  and  b=1 . So, the axis of symmetry is the vertical line

x=14x=−14

Intercepts

You can find the yy -intercept of a parabola simply by entering 00 for xx . If the equation is in standard form, then you can just take cc as theyy -intercept. For instance, in the above example:

y=2(0)2+(0)1=1y=2(0)2+(0)−1=−1

So the yy -intercept is 1−1 .

The xx -intercepts are a bit trickier. You can use factoring , or completing the square , or the quadratic formula to find these (if they exist!).

Domain and Range

As with any function, the domain of a quadratic function f(x)f(x) is the set of xx -values for which the function is defined, and the range is the set of all the output values (values of ff ).

Quadratic functions generally have the whole real line as their domain: any xx is a legitimate input. The range is restricted to those points greater than or equal to the yy -coordinate of the vertex (or less than or equal to, depending on whether the parabola opens up or down).

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