Question

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

Short answer

Answer: The end behavior of the function \(y=x^5\) is that as \(x \to \infty\), \(y \to \infty\), and as \(x \to -\infty\), \(y \to -\infty\). This means that the graph will rise to the right and fall to the left.

Step-by-step solution

Identify intercepts and symmetry

First, let's find x and y intercepts. For the x-intercept, set \(y=0\) and solve for \(x\): \(0 = x^5\) \(x = 0\) So, the x-intercept is at (0, 0). For the y-intercept, set \(x=0\) and solve for \(y\): \(y = 0^5\) \(y = 0\) So the y-intercept is also at (0, 0). Now, let's analyze the symmetry of the function. \(y=x^5\) is an odd-degree polynomial, and it is also an odd function because it satisfies the property \(f(-x) = -f(x)\). This means the graph will have rotational symmetry about the origin, so it will look similar in all four quadrants.

End behavior

To find the end behavior, we look at the behavior of the function as \(x\) approaches positive and negative infinity: - As \(x \to \infty\), \(y=x^5\to\infty\) - As \(x \to -\infty\), \(y=x^5\to-\infty\) This indicates that the graph will rise to the right and fall to the left.

Taking the first and second derivatives

Now, let's find the first and second derivatives of the function for analyzing the increasing/decreasing behavior and concavity: First derivative: \(y' = \frac{d}{dx}(x^5) = 5x^4\) Second derivative: \(y'' = \frac{d^2}{dx^2}(x^5) = 20x^3\)

Analyzing increasing/decreasing behavior and concavity

We will use the first derivative (\(y'\)) to determine where the function is increasing or decreasing: - Increasing: \(y' > 0 \Rightarrow 5x^4 > 0 \Rightarrow x \neq 0\) - Decreasing: \(y' < 0\), none So the graph of the function is always increasing except at \(x=0\). Now, let's analyze the concavity using the second derivative (\(y''\)): - Concave up: \(y'' > 0 \Rightarrow 20x^3 > 0 \Rightarrow x > 0\) - Concave down: \(y'' < 0 \Rightarrow 20x^3 < 0 \Rightarrow x < 0\) So the graph is concave up for \(x>0\) and concave down for \(x<0\).

Sketch the graph based on the analyzed information

1. Plot the intercepts at (0, 0). 2. Show odd function symmetry across quadrants. 3. Indicate that the graph increases as we move towards the right and decreases as we move towards the left. 4. Indicate the concave up behavior for \(x>0\) and concave down behavior for \(x<0\). With these key features, we finally have a complete sketch of the graph of the function \(y=x^5\).

Key concepts

Intercepts

In mathematics, intercepts are the points where a graph intersects the coordinate axes. Understanding intercepts gives us a useful insight into a polynomial graph's behavior and allows us to quickly identify significant points on a graph. Let's break down their significance:

• X-Intercept: This is where the graph touches the x-axis. To find it, set y to zero and solve for x. For the polynomial function \(y = x^5\), setting \(y = 0\) gives \(x = 0\). Thus, the x-intercept is at the origin (0, 0).
• Y-Intercept: This is where the graph intersects the y-axis. Set x to zero and calculate y. Again, for \(y = x^5\), \(y = 0^5 = 0\). Therefore, the y-intercept is also at (0, 0).

Knowing the intercepts offers the key points to start sketching a graph and to understand the initial behavior of the function.

Symmetry in Functions

Symmetry is a crucial property that helps describe the behavior of a polynomial graph. It tells us whether the graph will reflect across a certain line or remain the same when rotated around a point. Understanding the symmetry of a function assists us in simplifying graphing tasks and predicting the shape of functions. Let’s explore the symmetry in the context of polynomial functions like \(y = x^5\):

• Odd Functions: A function is considered odd if \(f(-x) = -f(x)\). The graph of \(y = x^5\) is an odd function, which means it exhibits rotational symmetry about the origin. This implies that if you rotate the graph by 180°, it will look the same.

Recognizing this symmetry makes it easier to sketch the graph by understanding that the pattern in one quadrant tells us about its behavior in the others.

End Behavior

The end behavior of a graph describes how the function behaves as \(x\) approaches positive or negative infinity. Knowing the end behavior helps us understand the trends of large and small values in a polynomial. Here's what you should note about \(y = x^5\):

• Positive Infinity: As \(x\) increases towards infinity, \(y = x^5\) also increases towards infinity. This indicates that the graph rises to the right.
• Negative Infinity: As \(x\) decreases towards negative infinity, \(y = x^5\) portrays a decreasing pattern, heading toward negative infinity. This pattern implies that the graph falls to the left.

In summary, understanding end behavior is vital in predicting the 'ends' of the graph, giving us a comprehensive view of how the function stretches beyond visible points on the chart.

Derivatives

Derivatives are a fundamental tool in calculus that help us determine the rate of change of a function. Through derivatives, we gain insight into a function's increasing/decreasing behavior and concavity.

• First Derivative \(y'\): This tells us where the graph is increasing or decreasing. For \(y = x^5\), the first derivative is \(y' = 5x^4\). Since \(5x^4 > 0\) for all \(x eq 0\), the graph is always increasing.
• Second Derivative \(y''\): This provides information about the concavity of the graph. For the function \(y = x^5\), the second derivative is \(y'' = 20x^3\). The graph is concave up for \(x > 0\) (where \(y'' > 0\)) and concave down for \(x < 0\) (where \(y'' < 0\)).

The derivatives allow us to construct a detailed and precise sketch of the graph by understanding how its curves and slopes change.