Problem 55
Question
Graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$f(x)=x^{3}-16 x$$
Step-by-Step Solution
Verified Answer
x-intercepts: \((-4, 0), (0, 0), (4, 0)\); y-intercept: \((0, 0)\); end behavior: right up, left down.
1Step 1: Identify the polynomial
The given polynomial function is \( f(x) = x^3 - 16x \). This is a cubic polynomial, which means its graph has an S-shaped curve and may have up to 3 roots or x-intercepts.
2Step 2: Calculate the x-intercepts
To find the x-intercepts, set \( f(x) = 0 \): \[x^3 - 16x = 0\]Factor the equation: \[x(x^2 - 16) = 0\]Set each factor to zero:1. \( x = 0 \)2. \( x^2 - 16 = 0 \) which further factors to \( (x - 4)(x + 4) = 0 \) leading to the solutions \( x = 4 \) and \( x = -4 \). Thus the x-intercepts are at \((0, 0)\), \((4, 0)\), and \((-4, 0)\).
3Step 3: Calculate the y-intercept
The y-intercept is found by evaluating the polynomial at \( x = 0 \):\[f(0) = 0^3 - 16(0) = 0\]Thus, the y-intercept is at the origin \((0, 0)\).
4Step 4: Determine the end behavior
For a cubic polynomial \( ax^3 \), the end behavior is determined by the leading term. As \( x \to \infty \), the term \( x^3 \) dominates and \( f(x) \to \infty \). As \( x \to -fty \), \( x^3 \to -fty \) so \( f(x) \to -fty \). This indicates the ends of the graph are going up on the right and down on the left.
5Step 5: Sketch and interpret the graph
Use a graphing calculator to graph the function \( f(x) = x^3 - 16x \). Verify that the graph crosses the x-axis at \((-4, 0)\), \((0, 0)\), and \((4, 0)\), confirming the intercept calculations. Observe that the graph's ends go up to the right and down to the left.
Key Concepts
Cubic PolynomialX-interceptsY-interceptEnd Behavior
Cubic Polynomial
A cubic polynomial is a type of polynomial function where the highest degree of any term is three. In our example, the function \( f(x) = x^3 - 16x \) is a cubic polynomial because it has the term \( x^3 \). This term, being of degree three, defines the overall shape of the graph.
A cubic polynomial is often graphically represented with an S-curve, which has a characteristic wave-like shape. This shape enables the curve to cross the x-axis up to three times, which affects both the number of intercepts and the roots of the function.
Recognizing a cubic polynomial is crucial in predicting the general behavior of the graph, such as its turning points and intercepts, especially since these functions can rise and fall multiple times before continuing towards infinity.
A cubic polynomial is often graphically represented with an S-curve, which has a characteristic wave-like shape. This shape enables the curve to cross the x-axis up to three times, which affects both the number of intercepts and the roots of the function.
Recognizing a cubic polynomial is crucial in predicting the general behavior of the graph, such as its turning points and intercepts, especially since these functions can rise and fall multiple times before continuing towards infinity.
X-intercepts
The x-intercepts of a polynomial function are points where the graph crosses the x-axis. To find these intercepts, we solve the equation \( f(x) = 0 \). In our function \( f(x) = x^3 - 16x \), this involves factoring the equation.
The factorization process reveals that \( x(x^2 - 16) = 0 \). Further breaking down \( x^2 - 16 \) results in \( (x - 4)(x + 4) = 0 \). From this result, the solutions or roots are \( x = 0 \), \( x = 4 \), and \( x = -4 \).
This means the x-intercepts are located at the points \((0, 0)\), \((4, 0)\), and \((-4, 0)\). Each intercept tells us a specific point where the function equals zero and visually indicates where the graph crosses the x-axis. Finding x-intercepts is essential for sketching the graph precisely and understanding the roots of the equation.
The factorization process reveals that \( x(x^2 - 16) = 0 \). Further breaking down \( x^2 - 16 \) results in \( (x - 4)(x + 4) = 0 \). From this result, the solutions or roots are \( x = 0 \), \( x = 4 \), and \( x = -4 \).
This means the x-intercepts are located at the points \((0, 0)\), \((4, 0)\), and \((-4, 0)\). Each intercept tells us a specific point where the function equals zero and visually indicates where the graph crosses the x-axis. Finding x-intercepts is essential for sketching the graph precisely and understanding the roots of the equation.
Y-intercept
The y-intercept of a function is where the graph intersects the y-axis. To determine this, we evaluate the function when \( x = 0 \). For the cubic function \( f(x) = x^3 - 16x \), substituting \( x = 0 \) gives \( f(0) = 0^3 - 16(0) = 0 \).
Thus, the y-intercept for the given polynomial is at the origin, \((0, 0)\). This point indicates where the function crosses the y-axis and provides a starting point for graphing the polynomial.
Understanding the y-intercept is crucial, as it helps align the graph vertically on a coordinate plane, serving as a reference point from which the function can be visualized and plotted.
Thus, the y-intercept for the given polynomial is at the origin, \((0, 0)\). This point indicates where the function crosses the y-axis and provides a starting point for graphing the polynomial.
Understanding the y-intercept is crucial, as it helps align the graph vertically on a coordinate plane, serving as a reference point from which the function can be visualized and plotted.
End Behavior
The end behavior of a polynomial function describes the direction of the graph as the value of \( x \) approaches infinity or negative infinity. For the cubic polynomial \( f(x) = x^3 - 16x \), the leading term \( x^3 \) plays a pivotal role in determining this behavior.
As \( x \to \infty \), the term \( x^3 \) becomes very large and positive, causing the entire function \( f(x) \) to trend upward towards infinity. Conversely, as \( x \to -\infty \), \( x^3 \) becomes very large and negative, causing \( f(x) \) to trend downward toward negative infinity.
This characteristic tells us that the graph of a cubic polynomial like \( f(x) = x^3 - 16x \) will rise to the right and fall to the left. Recognizing the end behavior is essential for properly sketching and understanding the full scope of the polynomial's graph.
As \( x \to \infty \), the term \( x^3 \) becomes very large and positive, causing the entire function \( f(x) \) to trend upward towards infinity. Conversely, as \( x \to -\infty \), \( x^3 \) becomes very large and negative, causing \( f(x) \) to trend downward toward negative infinity.
This characteristic tells us that the graph of a cubic polynomial like \( f(x) = x^3 - 16x \) will rise to the right and fall to the left. Recognizing the end behavior is essential for properly sketching and understanding the full scope of the polynomial's graph.
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