Maximum and Minimum Values A quadratic function is given. (a) Use a graphing device to find the

Maximum and Minimum Values A quadratic function is given....

Key Concepts

Quadratic Functions

A quadratic function is a second-degree polynomial of the form ax² + bx + c. Its graph is a parabola that opens upward if a is positive, indicating a minimum point, or downward if a is negative, indicating a maximum point. Understanding the general structure of quadratic functions is essential for analyzing their properties, such as symmetry, vertex, and axis of symmetry.

Vertex of a Parabola

The vertex of a parabola is the point where the quadratic function attains its maximum or minimum value. It is a crucial concept because it represents the optimal point of the function. The coordinates of the vertex can be found using the formula (-b/(2a), f(-b/(2a))) or by completing the square to convert the quadratic function into vertex form.

Graphical Analysis

Graphical analysis involves using graphing devices or software to visualize and approximate the behavior of functions. In the context of quadratic functions, it allows one to see the overall shape of the parabola and to estimate the location and value of the vertex. This method is particularly useful for obtaining a quick, approximate answer before diving into more precise algebraic calculations.

Exact Algebraic Techniques

Exact algebraic methods, such as completing the square or applying the vertex formula, provide a precise way of calculating the vertex of a quadratic function. These techniques transform the function into a form where the maximum or minimum value is clearly visible, ensuring that the optimal value can be determined without relying on approximations.

Rounding and Approximation

Rounding and approximation are important when numerical devices are used to evaluate functions or when answers are required in a specified level of precision, such as two decimal places. This concept ensures that the results are both practical for real-world applications and consistent with the levels of precision often required in mathematical contexts.

Transcript

00:01 Okay, so the first part of this question is asking us to find the, it's asking us to find the maximum minimum value of the function by using, so you can use a calculator or you can use a graphical device.

00:16 I'll just use my calculator.

00:19 I'm using the cassio fx 891 or 991 ex at the moment.

00:27 Right, so if you just type the equation into the, it's going to be different for every calculator, so it's hard for me to help you along with this right now but just ask your tutor or your teacher how to do this on your particular device maybe that you need to get a new device i'm not sure really but it's always good investing in a cap rate that can do this sort of thing so if you put this into your calculator what i get as a minimum value is i get minus four point and it asks us to round to two decimal places i get minus 4 .0 1 to 2 decimal places.

01:10 Okay.

01:11 And then part b actually to find the minimum minimum or maximum value without doing that.

01:18 And okay, so the way i'm going to find the maximum or minimum values, i'm actually going to use the maximum or minimum value equation to find the x coordinate as a vertex and then i'm going to sub that in.

01:28 So the x coordinate of the vertex is minus b over to a here.

01:34 And that's going to equal minus 1 .79 over two times by one so that's just over two and i put that into my calculator i don't want to be rounding at all here because that'll that i'll reduce the accuracy of my answer i get just minus 179 over 200 and i can just keep it as that for the moment and then i'll sub that back into the equation and i'll see what that gives me.

02:11 So minus 179 over 200 all squared plus 1 .79 times by 179 over 200 minus 3 .21...

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