Problem 63
Question
For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or -1 . There may be more than one correct answer. The \(y\) - intercept is (0,0) . The \(x\) - intercepts are \((0,0),(2,0) .\) Degree is \(3 .\) End behavior: as \(x \rightarrow-\infty, \quad f(x) \rightarrow-\infty,\) as \(x \rightarrow \infty, f(x) \rightarrow \infty\)
Step-by-Step Solution
Verified Answer
The function is \(f(x) = x^3 - 2x^2\).
1Step 1: Analyze the intercepts
The given intercepts are \(y\)-intercept: (0,0)\(x\)-intercepts: (0,0) and (2,0).Thus, the polynomial has roots at \(x=0\) and \(x=2\). Since (0,0) is both an \(x\)-intercept and a \(y\)-intercept, \(x=0\) must be a root with multiplicity.
2Step 2: Determine the roots and their multiplicities
The root at \(x=0\) is a repeated root since it is also the \(y\)-intercept.Therefore, let's assume the root \(x=0\) has a multiplicity of 2, and the root \(x=2\) has multiplicity 1 to get a degree 3 polynomial:\((x-0)^{2}(x-2)\).
3Step 3: Construct the polynomial function
Based on the roots and their multiplicities, the polynomial function is:\(f(x) = a(x^2)(x-2)\).
4Step 4: Use end behavior to determine the leading coefficient
The end behavior as \(x \to -\infty, f(x) \to -\infty\) and as \(x \to \infty, f(x) \to \infty\) indicates the leading coefficient, \(a\), should be positive. Hence, \(a = 1\).
5Step 5: Write the final polynomial
The final polynomial can be simplified as:\(f(x) = (x^2)(x-2) = x^3 - 2x^2\).
Key Concepts
InterceptsRoots and MultiplicitiesEnd BehaviorLeading Coefficient
Intercepts
Intercepts are the points where a graph crosses the axes. These points are vital because they give us clues about the roots of the polynomial and the nature of the polynomial itself.
The y-intercept is where the graph crosses the y-axis, meaning the point has an x-value of zero. For this exercise, the y-intercept is at (0, 0). This is where the graph starts from the y-axis.
On the other hand, x-intercepts are the points where the graph crosses the x-axis. These are the values of x for which the function returns zero. For this polynomial function, the x-intercepts are (0, 0) and (2, 0). This tells us that the polynomial has roots at these points.
The y-intercept is where the graph crosses the y-axis, meaning the point has an x-value of zero. For this exercise, the y-intercept is at (0, 0). This is where the graph starts from the y-axis.
On the other hand, x-intercepts are the points where the graph crosses the x-axis. These are the values of x for which the function returns zero. For this polynomial function, the x-intercepts are (0, 0) and (2, 0). This tells us that the polynomial has roots at these points.
- Y-intercept: where x = 0
- X-intercepts: where the function returns 0
Roots and Multiplicities
Roots of a polynomial are the solutions to the equation when the polynomial is set to zero. These are critical for understanding the polynomial's structure. In this exercise, the roots identified from the intercepts are 0 and 2.
A root's multiplicity refers to the number of times that a particular root occurs in the polynomial. A root with a higher multiplicity will touch or "bounce" off the axis but not cross it. If a root has a multiplicity of two, it will generally appear as a turning point in the graph.
For the given polynomial function, the root at x = 0 appears with multiplicity 2 since it is also a y-intercept. This means the graph touches the x-axis but turns back the way it came. On the contrary, the root at x = 2 has a multiplicity of 1, and the graph will cross the x-axis here.
A root's multiplicity refers to the number of times that a particular root occurs in the polynomial. A root with a higher multiplicity will touch or "bounce" off the axis but not cross it. If a root has a multiplicity of two, it will generally appear as a turning point in the graph.
For the given polynomial function, the root at x = 0 appears with multiplicity 2 since it is also a y-intercept. This means the graph touches the x-axis but turns back the way it came. On the contrary, the root at x = 2 has a multiplicity of 1, and the graph will cross the x-axis here.
- Multiplicity of 2 for x = 0
- Multiplicity of 1 for x = 2
End Behavior
End behavior describes how a polynomial behaves as x approaches positive or negative infinity. This is crucial in determining the overall shape of the polynomial and helps us deduce the sign of the leading coefficient.
The exercise describes the end behavior as: when x goes to negative infinity, f(x) goes to negative infinity, and when x moves towards positive infinity, f(x) also heads towards positive infinity. This type of behavior suggests a cubic polynomial with a positive leading coefficient.
The fact that the function behaves this way is directly linked to its odd degree. The polynomial's highest degree term, which is positive, dominates the end behavior as x grows larger.
The exercise describes the end behavior as: when x goes to negative infinity, f(x) goes to negative infinity, and when x moves towards positive infinity, f(x) also heads towards positive infinity. This type of behavior suggests a cubic polynomial with a positive leading coefficient.
The fact that the function behaves this way is directly linked to its odd degree. The polynomial's highest degree term, which is positive, dominates the end behavior as x grows larger.
- Negative endless extension: x → -∞, f(x) → -∞
- Positive endless extension: x → ∞, f(x) → ∞
Leading Coefficient
The leading coefficient is paramount as it influences the direction of the end behavior of a polynomial function. It's the coefficient of the term with the highest degree. For our polynomial function, this coefficient determines whether the graph will rise or fall as it extends into infinite directions.
In the given exercise, the end behavior indicates that the graph decreases as x goes toward -∞ and increases as x goes toward +∞. This behavior implies that the leading coefficient must be positive. Here, assuming the degree of the term is 3, a positive leading coefficient suggests the simplest form for this polynomial is with a leading coefficient of 1.
Thus, the final polynomial becomes f(x) = x³ - 2x², confirming that the leading coefficient is indeed positive and matches the end behavior requirement.
In the given exercise, the end behavior indicates that the graph decreases as x goes toward -∞ and increases as x goes toward +∞. This behavior implies that the leading coefficient must be positive. Here, assuming the degree of the term is 3, a positive leading coefficient suggests the simplest form for this polynomial is with a leading coefficient of 1.
Thus, the final polynomial becomes f(x) = x³ - 2x², confirming that the leading coefficient is indeed positive and matches the end behavior requirement.
- Determines end behavior
- Positive for the given problem
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