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

Answer: The graphs of both functions have the same key points at (-1, 1), (0, 0), and (1, 1), and they both approach positive infinity as x approaches positive or negative infinity due to their even powers. However, the graph of $$y = x^6$$ is steeper than the graph of $$y = x^4$$, especially for larger x values.

Step-by-step solution

Identify Key Points

We will start by finding the values of the functions at some key points such as x = -1, x = 0, and x = 1. For the first function, when x = -1, x = 0, and x = 1: $$y = (-1)^4 = 1$$ $$y = (0)^4 = 0$$ $$y = (1)^4 = 1$$ For the second function, when x = -1, x = 0, and x = 1: $$y = (-1)^6 = 1$$ $$y = (0)^6 = 0$$ $$y = (1)^6 = 1$$

Analyze End Behavior

Next, analyze the behavior of both functions as x approaches positive or negative infinity by looking at the power of x: - As x approaches positive infinity: - $$y = x^4$$ → y also approaches positive infinity - $$y = x^6$$ → y also approaches positive infinity - As x approaches negative infinity: - $$y = x^4$$ → y approaches positive infinity (because of the even power) - $$y = x^6$$ → y approaches positive infinity (also because of the even power)

Draw the Graphs

Start by drawing the coordinate plane and plot the key points for both functions: - For $$y = x^4$$, plot the points (-1, 1), (0, 0), and (1, 1). - For $$y = x^6$$, plot the points (-1, 1), (0, 0), and (1, 1). Next, connect the points for each function by considering their end behavior. - For $$y = x^4$$, the graph will be a smooth curve that goes through the plotted points and tends to positive infinity as x approaches positive or negative infinity. - For $$y = x^6$$, the graph will also be a smooth curve that goes through the plotted points and tends to positive infinity as x approaches positive or negative infinity. Since both functions have an even power, they will be symmetric about the y-axis. Make sure the graph of $$y=x^6$$ is steeper than that of $$y=x^4$$, especially for larger x values.

Compare the Graphs

Finally, compare the graphs of the two functions. Note that they intersect at three points: (-1, 1), (0, 0), and (1, 1). Also, observe that the graph of $$y = x^6$$ is steeper than the graph of $$y = x^4$$, particularly as the x values get larger.