X3-x2 Graph

[Nathen Paisley]

Wellsince we can't actually generate plots of things that have more than 3 dimensions, it will be challenging to try and show larger dimensional DOEs, but I wouldn't consider a 4-factor DOE to be a tesseract because the three axes that make up the cube are three of the four factors in the DOE design space, so the extra cube in the cube is still in the DOE design space of those three dimensions.

In my opinion, especially when explaining DOEs to non-DOE experts, it's actually better to keep things as a 2D plot. So, if you have 3 factors, you'd plot X1 vs X2, X1 vs X3, and X2 vs X3. Similarly for 4 factors, it would be X1/X2, X1/X3, X1/X4, X2/X3, X2/X4, X3/X4, and so on for n-factor DOEs. You can see that as the number of factors increases, the number of plots goes up pretty quick. Most people often find two dimensional graphs to be more understandable, and sometimes 3D scatterplots can be a bit confusing and overwhelming for a viewer, especially when the graph is static and the person can't turn the scatterplot. 2D just makes things a lot easier to understand. The good thing is that with Graph Builder, you can put several different columns (factors) on either the X-axis or Y-axis, so you can have multiple plots in a single graphic.

Local maximum and minimum points are quite distinctive on the graph ofa function, and are therefore useful in understanding the shape of thegraph. In many applied problems we want to find the largest orsmallest value that a function achieves (for example, we might wantto find the minimum cost at which some task can be performed) and soidentifying maximum and minimum points will be useful for appliedproblems as well. Some examples of local maximum and minimum pointsare shown in figure 5.1.1.

Since the derivative is zero or undefined at both local maximum andlocal minimum points, we need a way to determine which, if either,actually occurs. The mostelementary approach, but one that is often tedious or difficult, is totest directly whether the $y$ coordinates "near'' the potentialmaximum or minimum are above or below the $y$ coordinate at the pointof interest. Of course, there are too many points "near'' the pointto test, but a little thought shows we need only test two provided weknow that $f$ is continuous (recall that this means that the graph of$f$ has no jumps or gaps).

The graphs of rational functions are characterized by asymptotes. Asymptotes are lines that the curve approaches at the edges of the coordinate plane. Vertical asymptotes occur where the denominator of a rational function approaches zero. A rational function cannot cross a vertical asymptote because it would be dividing by zero. Horizontal asymptotes occur when the x-values get very large in the positive or negative direction. Horizontal asymptotes can be crossed. Slant asymptotes are similar to horizontal asymptotes but are slanted lines.

The domain of a rational function cannot include a value that makes the denominator equal zero because that causes the function to be undefined. To find the domain of a rational function, set the denominator equal to zero and solve for x.

Sometimes a graph of a rational function will contain a hole. A hole is a single point where the graph is not defined and is indicated by an open circle. These holes come from the factors of the denominator that cancel with a factor of the numerator. When the function is simplified, the hole disappears. Thus, these types of holes are called removable discontinuities.

Notice that a graph of a rational function will never cross a vertical asymptote, but the graph may cross a horizontal or slant asymptote. Also, the graph of a rational function may have several vertical asymptotes, but the graph will have at most one horizontal or slant asymptote.

In general, if the degree of the numerator is larger than the degree of the denominator, the end behavior of the graph will be the same as the end behavior of the quotient of the rational fraction. For example, the function \(f(x) = \fracx^32x - 2\) will have end behavior like a quadratic function because the quotient is a quadratic with the function is divided.

In this case the end behavior is [latex]f\left(x\right)\approx \frac4xx^2=\frac4x[/latex]. This tells us that, as the inputs increase or decrease without bound, this function will behave similarly to the function [latex]g\left(x\right)=\frac4x[/latex], and the outputs will approach zero, resulting in a horizontal asymptote at [latex]y=0[/latex]. Note that this graph crosses the horizontal asymptote.

To find the equation of the slant asymptote, divide [latex]\dfrac3x^2-2x+1x - 1[/latex]. The quotient is [latex]3x+1[/latex], and the remainder is 2. The slant asymptote is the graph of the line [latex]g\left(x\right)=3x+1[/latex].

Case 3: If the degree of the denominator = degree of the numerator, there is a horizontal asymptote at [latex]y=\fraca_nb_n[/latex], where [latex]a_n[/latex] and [latex]b_n[/latex] are the leading coefficients of [latex]p\left(x\right)[/latex] and [latex]q\left(x\right)[/latex] for [latex]f\left(x\right)=\fracp\left(x\right)q\left(x\right),q\left(x\right)\ne 0[/latex].

Notice that, while the graph of a rational function will never cross a vertical asymptote, the graph may or may not cross a horizontal or slant asymptote. Also, although the graph of a rational function may have many vertical asymptotes, the graph will have at most one horizontal (or slant) asymptote.

It should be noted that, if the degree of the numerator is larger than the degree of the denominator by more than one, the end behavior of the graph will mimic the behavior of the reduced end behavior fraction. For instance, if we had the function

Both the numerator and denominator are linear (degree 1). Because the degrees are equal, there will be a horizontal asymptote at the ratio of the leading coefficients. In the numerator, the leading term is [latex]t[/latex], with coefficient 1. In the denominator, the leading term is [latex]10t[/latex], with coefficient 10. The horizontal asymptote will be at the ratio of these values:

This tells us that as the values of [latex]t[/latex] increase, the values of [latex]C[/latex] will approach [latex]\frac110[/latex]. In context, this means that, as more time goes by, the concentration of sugar in the tank will approach one-tenth of a pound of sugar per gallon of water or [latex]\frac110[/latex] pounds per gallon.

The function will have vertical asymptotes when the denominator is zero, causing the function to be undefined. The denominator will be zero at [latex]x=1,-2,\textand 5[/latex], indicating vertical asymptotes at these values.

The numerator has degree 2, while the denominator has degree 3. Since the degree of the denominator is greater than the degree of the numerator, the denominator will grow faster than the numerator, causing the outputs to tend towards zero as the inputs get large, and so as [latex]x\to \pm \infty , f\left(x\right)\to 0[/latex]. This function will have a horizontal asymptote at [latex]y=0[/latex].

A rational function will have a y-intercept when the input is zero, if the function is defined at zero. A rational function will not have a [latex]y[/latex]-intercept if the function is not defined at zero.

Likewise, a rational function will have [latex]x[/latex]-intercepts at the inputs that cause the output to be zero. Since a fraction is only equal to zero when the numerator is zero, [latex]x[/latex]-intercepts can only occur when the numerator of the rational function is equal to zero.

Because the numerator is the same degree as the denominator we know that as [latex]x\to \pm \infty , f\left(x\right)\to -4; \textso y=-4[/latex] is the horizontal asymptote. Next, we set the denominator equal to zero, and find that the vertical asymptote is [latex]x=3[/latex], because as [latex]x\to 3,f\left(x\right)\to \infty[/latex]. We then set the numerator equal to 0 and find the [latex]x[/latex]-intercepts are at [latex]\left(2.5,0\right)[/latex] and [latex]\left(3.5,0\right)[/latex]. Finally, we evaluate the function at 0 and find the [latex]y[/latex]-intercept to be at [latex]\left(0,\frac-359\right)[/latex].

The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downwardand vary in "width" or "steepness", but they all have the same basic "U" shape. Thepicture below shows three graphs, and they are all parabolas.

You know that two points determine a line. This means that if you are given any two points in the plane, thenthere is one and only one line that contains both points. A similar statement can be made about points and quadraticfunctions.

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