Question

Make a sketch of the given pairs of functions. Be sure to draw the graphs accurately relative to each other. $$y=x^{4} \text { and } y=x^{6}$$

Short answer

The main difference in the shapes of the graphs of the functions \(y=x^4\) and \(y=x^6\) is that the graph of \(y=x^6\) is flatter near the origin and grows more rapidly as x moves away from 0, while the graph of \(y=x^4\) is steeper near the origin and becomes less steep as x moves away from the origin. Both graphs are symmetric about the y-axis and share the minimum point at the origin (0, 0).

Step-by-step solution

Understand the shape of each function

The given functions are both polynomial functions and are even-degree functions. Polynomials are continuous and smooth curves. In general, for positive even-degree polynomials, the graph extends to infinity in both the positive and negative y-directions. Also, the behavior of the graph for positive x values will be similar to that of the negative x values since their corresponding y-values will be identical due to the even exponent. Let's now discuss the shape of each given function.

Sketch \(y=x^4\)

The function \(y=x^4\) has its minimum point at the origin (0, 0), and the graph extends to infinity as x approaches both positive and negative infinity. The curve is symmetric about the y-axis and looks like a steeper version of a parabola. To sketch the graph of \(y=x^4\), start at the origin and draw the curve gradually moving away from the y-axis, extending towards infinity. Make sure to sketch the graph symmetrically on both sides of the y-axis.

Sketch \(y=x^6\)

The function \(y=x^6\) also has its minimum point at the origin (0, 0) and the graph extends to infinity as x approaches both positive and negative infinity. Again, the curve is symmetric about the y-axis. However, \(y=x^6\) is "flatter" than \(y=x^4\) near the origin, but it grows much more rapidly as x moves away from 0. To sketch the graph of \(y=x^6\), start at the origin and draw the curve very flat initially, then let it grow more rapidly than the \(y=x^4\) curve as x moves away from the origin. Again, make sure to sketch the graph symmetrically on both sides of the y-axis.

Identify any intersections

In this case, since both functions have minimum points at the origin, the two graphs clearly intersect at (0,0). There are no other points of intersection in this case.

Sketch the graphs accurately relative to each other

Now that we understand the shape of each function and identified their intersection, we can sketch an accurate representation of both graphs relative to each other. Start by sketching the graph of \(y=x^4\), using the information from Step 2. Then, sketch the graph of \(y=x^6\), using the information from Step 3, making sure that it is flatter near the origin and its growth is more rapid as x moves away from 0. Remember that the graphs have different slopes, with the \(y=x^6\) curve having initially a gentler slope near the origin and a steeper slope away from the origin. Ensure that both graphs intersect at the origin (0,0).

Key concepts

Even-Degree Polynomials

Even-degree polynomials, like the ones in our exercise, have several notable characteristics. First, they are symmetrical about the y-axis, meaning that the left side of the graph is a mirror image of the right side. This occurs because the even exponent produces the same y-value for both positive and negative x-values. For example, with the function y=x^4, if you input x=2 or x=-2, you'd get the same result: y=16.

Moreover, these functions do not have any sharp turns or cusps; they are smooth and continuous. As we go towards infinity in the x-direction, the y-value of the graph will tend to infinity as well. If the leading coefficient (the number before the variable) is positive, the ends of the graph will point upwards; if it's negative, they would both point downwards. In the case of the functions y=x^4 and y=x^6, since both have a positive coefficient of 1, their end behaviors will extend upwards indefinitely.

Graph Symmetry

Understanding graph symmetry is pivotal when sketching polynomial functions. As mentioned, even-degree polynomials exhibit symmetry about the y-axis—this is known as y-axis symmetry. In technical terms, a function has y-axis symmetry if and only if f(x) = f(-x) for all x in the domain of the function.

This property can significantly simplify sketching and analysis because you only need to calculate the function values for the positive x-values and then reflect these points across the y-axis to find the corresponding negative x-values. For our functions y=x^4 and y=x^6, this means that after plotting the curve on the right half of the graph, you can perform a 'mirror image' to achieve the left half of the graph. This symmetry also tells us that the graphs will have no shifts or tilts to either side, maintaining a balanced shape centered on the y-axis.

Polynomial Intersections

The points where two or more polynomial graphs intersect are known as their points of intersection. Finding these points is crucial when graphing multiple functions on the same set of axes, as it helps us understand the relationship between the functions. To find the intersection, one would typically set the functions equal to each other and solve for x.

In our exercise, the functions y=x^4 and y=x^6 intersect at the origin (0,0). We were able to determine this quickly because both functions pass through the origin. The exercise does not require solving for intersections because there are no other points where the two graphs meet. It is important to note that the number and location of intersections can reveal much about the behavior of the functions involved, including their relative rates of growth or decay as x-values increase or decrease.