In Exercises 17–18, use the function’s equation, and not its graph, to find a. the minimum or ma

In Exercises 17–18, use the function’s equation, and not its...

Key Concepts

Quadratic Functions

Quadratic functions are expressions of the form ax² + bx + c, where a, b, and c are constants and a is nonzero. They are fundamental in algebra and are known for producing parabolic graphs. Understanding quadratic functions is crucial because they appear in many areas of mathematics and science, and their properties, such as symmetry, vertex, and direction of opening, are essential for analysis and problem solving.

Vertex of a Parabola

The vertex of a parabola is the point where the function attains its extremum (minimum or maximum value). It is a key feature in quadratic functions that indicates the turning point of the graph. The vertex can be found using the formula x = -b/(2a) for the x-coordinate, followed by substitution into the original function to obtain the y-coordinate. This concept is pivotal in optimization problems and in understanding the behavior of quadratic graphs.

Completing the Square

Completing the square is a technique used to rewrite a quadratic function into a form that clearly reveals its vertex and makes analysis easier. By transforming the quadratic expression into vertex form, one can directly identify the minimum or maximum value of the function. This method is widely used to solve quadratic equations, analyze parabolic motion, and determine function properties such as the vertex and the axis of symmetry.

Domain and Range

The domain of a function includes all the input values for which the function is defined, while the range consists of the output values that result from these inputs. In the case of quadratic functions, the domain is typically all real numbers because any real number can be substituted for x, but the range is dependent on the vertex and the orientation of the parabola. Understanding the domain and range is essential in function analysis, ensuring that the behavior of the function is completely characterized.

Transcript

00:01 Okay, we want to answer three questions about this equation and we want to do it using the equation and not graphing it but just so you can sort of see our equations are x squared so it's going to look something like this and we're going to have to try and find a minimum but we're going to do it using only the equation and not graphing it at all okay, so finding a maximum minimum the interesting property about it is that the gradient is zero okay, so to find our maximum min we use simply and to find a gradient obviously you just need to differentiate this equation.

00:34 So to find our maximum min we'll just differentiate our equation and equate it to zero and find out what our x values could be.

00:41 Okay, so differentiation take the power minus one times about what that power was at the front so two times two is four minus one it's just going to be x so you've got four x.

00:52 Here you've got x to the one minus one it becomes x to the zero so that just becomes one, 12 times 1 is 12 and we want to create to the zero to find our maximum minimum.

01:02 So you'll find bringing 12 across divide by 4, you'll have x is equal to minus 3.

01:09 Okay, so our maximum minimum point is going to be minus 3 and to find out if it's the maximum or minimum all you have to do is take the second derivative and if that is above zero it's a minimum and if it's below zero it's a sorry if it's above zero it's a minimum and if it's below zero it's a maximum so we're just going to be left with four.

01:33 I hope you can see that.

01:34 This is going to turn to nothing.

01:35 This will turn to 1 times 4.

01:39 So we know that this is a minimum...

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