A quadratic function f is given. a. Express f in standard form. b. Sketch a graph of f c. Find t

A quadratic function f is given. a. Express f in standard...

Key Concepts

Quadratic Function

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants. This function produces a parabolic graph, which opens upward if the coefficient a is positive and downward if a is negative. Quadratic functions have key features such as a vertex, an axis of symmetry, and sometimes a maximum or minimum value.

Standard (Vertex) Form

The standard or vertex form of a quadratic function is written as f(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola. This form is particularly useful for quickly identifying the vertex and understanding the transformation of the graph relative to the parent function. Converting a quadratic from general form to vertex form involves the process of completing the square.

Completing the Square

Completing the square is a method used to transform a quadratic expression from its general form into the vertex form. The process involves manipulating the equation to form a perfect square trinomial plus a constant, which then directly reveals the vertex of the parabola. This technique is essential not only for rewriting the quadratic but also for analyzing the function's properties and graph.

Graph of a Quadratic Function

Graphing a quadratic function involves plotting a parabola based on its vertex, axis of symmetry, and intercepts. By expressing the quadratic in vertex form, one can easily determine the vertex, while the coefficient 'a' indicates the direction (upward or downward) and the width of the parabola. Additional points can be calculated by selecting x-values, ensuring an accurate sketch of the graph.

Maximum or Minimum Value

The maximum or minimum value of a quadratic function is found at its vertex. When the coefficient a is positive, the vertex represents the minimum point of the parabola; when a is negative, the vertex is the maximum point. This value is crucial for optimization problems and provides insight into the behavior of the function across its domain.

Transcript

00:01 Hello everyone in this question you have to find out part a, part 3 for this given function as an effect so if you consider part a what you have to find you have to convert it into standard form so what i'm doing here i'm taking 4 common minus 4 i'm taking common here then i'll have here x squared plus x and this plus 3 is same so further if i simplify then minus 4 here i'm doing x squared plus x squared plus 8.

00:32 X plus 1 by 4 minus 1 by 4 plus 3 so h of x equal to i can write finally minus 4 into this is x plus 1 by 2 all squared and here minus 4 into minus 1 by 4 that is 1 plus 1 plus 3 that is 4 so this is extended form for this given function next we have to pull over the graph so to pull out the graph what we have to do first we have to find why indescent which is h of 0 equal to if i take here so that is 3 next x intercept i have to find x intercept.

01:15 So how will find x intercept? h of x equal to i'll put 0.

01:19 So h of x equal to be already calculated which is minus 4 into x plus 1 by 2 whole square plus 4.

01:28 So this 4 if i take this side, so finally i'll get x plus 1 by 2 equal to square which is equal to 1 i'll get.

01:42 So if you take square root, so x plus 1 by 2 equal to i'll get.

01:46 Plus minus 1 it means x equal to i have minus 1 by 2 minus 1 and minus 1 by 2 plus 1 so 1 value is minus 3 by 2 another value is plus 1 by 2 so now if you pull out the graph so i'll get graph like this this is reading coefficient negative here so it will be open downward parabola and it is chipped 4 unit upward and half unit left side so at minus 1 by 2 will have peak point so at minus 1 by 2 is we check then we have value 4 so this is 4 over here and this is vertex and y intercepts are 1 is minus 3 by 2 another is 1 by 2 so the graph if is flawed then it will be like that so this is the graph and this point is y intercept which is 3 and this is origin 0 and let's say this is xx this is y xxxxx so this is the graph for this given function h of x...

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