Question

Sketch a graph of \(y=x^{5}\).

Short answer

Answer: The key features of the function \(y = x^5\) include its domain and range, which are both all real numbers; its y-intercept and x-intercept both at the point \((0,0)\); and its overall shape, resembling an "S". These features are used to sketch the graph by first plotting the intercepts and then drawing a smooth curve that enters the graph in quadrant II and exits in quadrant IV, passing through the origin.

Step-by-step solution

Determine the domain and range

The domain of the function \(y=x^5\) is all real numbers, which means that \(x\) can be any real number. The range of the function is also all real numbers, which means that \(y\) can take any real value.

Determine the y-intercept

To find the y-intercept, we set \(x=0\) and find the corresponding \(y\) value. \(y = (0)^5 = 0\). So, the y-intercept is at the point \((0,0)\).

Determine the x-intercept

To find the x-intercept, we set \(y = 0\) and solve for \(x\). \(0 = x^5 \Rightarrow x=0\). So, the x-intercept is also at the point \((0,0)\).

Determine the overall shape of the graph

Since the function \(y = x^5\) is an odd-degree polynomial with a positive leading coefficient, the graph will have a similar shape to \(y=x^3\). The graph will enter and exit opposite quadrants, which are quadrant II to quadrant IV.

Sketch the graph

Begin by plotting the x and y-intercepts at the point \((0,0)\). Then, sketch a smooth curve that demonstrates the overall shape described in Step 4. The graph enters along the line \(y=-x^5\) in quadrant II, passes through the origin \((0,0)\), and exits along the line \(y=x^5\) in quadrant IV. The final graph should have a distinctive "S" shape.

Key concepts

Understanding the Domain and Range of Polynomial Functions

For the function \(y = x^5\), determining the domain and range is the first step in graphing. The domain refers to all possible \(x\) values that can be input into the function. For any polynomial, the domain is typically all real numbers. This means \(x\) can take on any real value from negative infinity to positive infinity.

The range, on the other hand, is all possible \(y\) values that the function can output. Since \(y = x^5\) is a polynomial with an odd degree, it can produce every real number as \(y\). Therefore, the range is also all real numbers. This characteristic allows these functions to cover both ends of the vertical axis indefinitely.

Identifying X-Intercepts and Y-Intercepts

Intercepts are critical points on a graph where it crosses the axes.

• Y-Intercept: To find the y-intercept, set \(x = 0\) and solve for \(y\). In the case of \(y = x^5\), plugging in zero gives \(y = 0^5 = 0\), making the y-intercept at the origin, or point \((0,0)\).
• X-Intercept: To find the x-intercept, set \(y = 0\) and solve for \(x\). Since \(0 = x^5\) only holds true when \(x = 0\), the x-intercept is also at the origin, \((0,0)\).

These intercepts serve as key anchor points to begin sketching the polynomial's graph.

Characteristics of Odd-Degree Polynomials

An important feature of polynomial functions like \(y = x^5\) is their degree, which in this case is odd. Odd-degree polynomials have distinct graphing properties.

• They enter and exit the graph in opposite quadrants. For \(y = x^5\), this means it starts from the lower left (quadrant III) or upper left (quadrant II) and finishes in the upper right (quadrant I) or lower right (quadrant IV).
• They typically have a central symmetry about the origin, giving them an "S" shaped curve.

The leading coefficient is positive, ensuring that the graph of \(y = x^5\) will resemble an elongated "S," similar to the simpler \(y = x^3\).

Techniques for Sketching Graphs of Polynomials

To sketch \(y = x^5\), start by plotting the key intercepts. Here, it's the origin \((0,0)\).

With an understanding of the general behavior of odd-degree polynomials, begin from quadrant II, passing through the origin, and head toward quadrant IV.

Since there is no turning point or asymptote, focus on creating a smooth, continuous curve between these points. The curve's shape reflects the typical "S" pattern of odd-degree functions. Remember:

• Check the graph symmetry around the origin.
• Ensure it smoothly transitions from quadrant II to IV.

By keeping these characteristics in mind, you can accurately visualize and sketch polynomial graphs like \(y = x^5\) efficiently.