Find two quadratic functions, one that opens upward and one that opens downward, whose graphs ha

Find two quadratic functions, one that opens upward and one...

Key Concepts

Leading Coefficient and Parabola Orientation

The leading coefficient, a, of a quadratic function determines the direction in which the parabola opens. If a is positive, the parabola opens upward; if a is negative, it opens downward. This property is critical when constructing a quadratic function to meet specific conditions regarding its graph’s orientation.

Scaling Factor in Quadratic Functions

The constant 'a' in the factored form of a quadratic function acts as a scaling factor that adjusts the width and direction of the parabola. Any nonzero value of a will preserve the x-intercepts while affecting the graph's curvature and whether it opens upward or downward. This flexibility allows for multiple quadratic functions with the same zeros but different shapes.

X-intercepts (Roots or Zeros)

X-intercepts of a quadratic function are the points where its graph crosses the x-axis. These correspond to the solutions of the equation ax² + bx + c = 0. Knowing the x-intercepts helps in constructing the function by providing its factors since if r and s are roots, then the quadratic can be expressed as a(x - r)(x - s).

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically written in the form ax² + bx + c, where a, b, and c are constants and a is nonzero. These functions graph as parabolas and are fundamental in algebra and calculus, illustrating concepts such as symmetry, vertex, and axis of symmetry.

Factored Form of Quadratic Functions

The factored form of a quadratic function is derived from its x-intercepts and is written as a(x - r)(x - s), where r and s are the roots of the equation. This form is particularly useful for identifying the x-intercepts directly and for understanding how the function's graph is shaped by its zeros.

Transcript

00:01 Hello everyone in this question we have to find out two quality functions such that given x intercept and these two functions one should be open upward one should be open downward so what we can know here two x intercepts are given then we can find x coordinate of of vertex which is equal to average of these two are intercepts so average is minus 3 minus 1 by 2 divided by 2 which is equal to if i find then minus 7 by 4.

00:39 So if i write the equation then if vertex is non then equation we can write as a into x minus s all squared plus k where s comma k is the vertex where h is x coordinate so as coordinate we already know so if we place the value then we'll get a into x plus 7 by 4 all squared plus k over here now this quality function is passing through these two points so let's put any one point so we can let zero equal to a into yet actually if i put minus 3 so minus 3 plus 7 by 4 only square plus k now if i simplify this one what i'll get so i'll get this is equal to zero equal to a into this one if is upright so minus 3 so it can multiply so minus 12 plus 7 divided by 4 so minus 12 plus 7 is minus 5 by 4 so minus 5 by 4 if you do then i'll get 25 divided by 16 plus k so from here if i find the k value so k equal to i'll get minus 25 a divided by 16 now this k value if i place here in this function then quality function a for a i'll get as a into x plus 7 by 4 all squared minus 25a divided by 16 now one you have to find out open upward one is open downward so for open upward this problem should be positive so what i'll do i'll put a equal to 1 so equal to 1 if i put then f of x equal to i'll get x plus 7 by 4 whole squared minus 25 divided by 16 so this is open upward quality function next is open downward you have to find out so i'll put a equal to minus 1 then i'll get f of x equal to here minus 1 here also minus 1 so i'll get minus of x plus 7 by 4 polysquare plus 25 d '16.

02:49 So these are two functions...

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