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

The leading coefficient, denoted as 'a' in the quadratic function, determines the width of the parabola and the direction in which it opens. A positive leading coefficient results in a parabola that opens upward, while a negative leading coefficient yields a parabola that opens downward.

Factored Form of a Quadratic Function

When the x-intercepts (roots) of a quadratic are known, the quadratic can be expressed in the form a*(x - r1)*(x - r2), where r1 and r2 are the intercepts. This form directly shows the roots and makes it easier to manipulate or understand the function's behavior based on the roots.

Graphing Parabolas with Given Intercepts

Given the x-intercepts, one can derive quadratic functions in factored form and then choose an appropriate leading coefficient to ensure the desired opening direction (upward or downward). This method helps in constructing specific quadratic functions that satisfy certain graphical conditions.

Quadratic Function

A quadratic function is a second-degree polynomial typically expressed in the form ax² + bx + c. Its graph is a parabola whose shape and position are determined by the coefficients, and its evaluation at various x-values gives a characteristic U-shape or inverted U-shape depending on the leading coefficient.

X-Intercept (Root) of a Quadratic

The x-intercepts of a quadratic function are the points where the graph crosses the x-axis, meaning the function's output is zero at these points. These intercepts are the solutions to the equation ax² + bx + c = 0, and knowing them allows one to write the function in its factored form.

Transcript

00:01 Hello everyone in this question we have to find out two quality functions for given to x intercept one we have to find out up and downward another one is open upward so what i can say x coordinate of x coordinate of vertex we can write as x coordinate will be average of these two x intercept so i can write minus 5 by 2 plus 2 divided by 2 which is equal to i can write minus 1 1 by 4 so this is x coordinate now we can find the equation of quality function as fophe equal to a into x minus h whole square plus k where h comma k is the vertex so h is the x coordinate of vertex so here we know x coordinate of vertex is minus 1 by 4 so if i place here then i'll get equal to a into x plus 1 by 4 holly square plus k now this one is passing through these two points so i'll put any one point so let's put in 2 .0 if i put here then i'll get 0 equal to a into x value is 2 so 2 plus 1 by 4 which is 9 by 4 whole square plus k next if i simplify this one what i'll get 0 equal to this is 9 into not 9 a into 81 a into 81 divided by 60 16 plus k so if i simplify then k equal to i'll get minus 81a divided by 16 so k value if i place here then function i'll get f of x a into x plus 1 by 4 whole squared minus 81 divided by 16a so now you have to find out any two points uh two functions because here we can check this function depends on the value of a so we'll get so many functions so we have to find any 2.

02:20 1 is open upward, 1 is open downward.

02:22 So simply i am placing value of a equal to 1.

02:24 So if i place a equal to 1, then i'll get function equal to a is 1, then this is x plus 1 by 4, all square minus 1 divided by 16.

02:36 Next, what i can do, a equal to minus 1 i can put here.

02:41 So if i put minus 1, then f of h equal to i'll get a minus 1...

Please give Ace some feedback