Problem 47
Question
For the following exercises, graph the polynomial functions. Note \(x\) - and \(y\) - intercepts, multiplicity, and end behavior. $$ n(x)=-3 x(x+2)(x-4) $$
Step-by-Step Solution
Verified Answer
Intercepts: \((0,0), (-2,0), (4,0)\); end behavior: up on left, down on right.
1Step 1: Identify the Degree and Leading Coefficient
The given polynomial is \( n(x) = -3x(x+2)(x-4) \). First, expand it to identify the degree and leading coefficient. The polynomial in expanded form is \( -3x^3 - 6x^2 + 12x \). The degree of the polynomial is 3 (from \(x^3\)), and the leading coefficient is -3.
2Step 2: Identify x-Intercepts and Multiplicity
Set \(n(x) = 0\) to find the \(x\)-intercepts. This occurs at \(x = 0, x = -2,\) and \(x = 4\). Each factor is linear, so each intercept has a multiplicity of 1.
3Step 3: Determine the y-Intercept
To find the \(y\)-intercept, evaluate \(n(x)\) at \(x = 0\). Plugging in, \(n(0) = -3(0)(0+2)(0-4) = 0\). Therefore, the \(y\)-intercept is \((0,0)\).
4Step 4: Analyze End Behavior
Since the polynomial degree is odd and the leading coefficient is negative, the end behavior is: As \(x \to -\infty\), \(n(x) \to \infty\); and as \(x \to \infty\), \(n(x) \to -\infty\).
5Step 5: Sketch the Graph
Plot the intercepts: \((0, 0), (-2, 0), (4, 0)\). Since all intercepts are of multiplicity 1, the graph crosses at each intercept. For the end behavior, ensure the graph goes up to the left and down to the right. Sketch a smooth curve through the intercepts respecting this behavior.
Key Concepts
Degree of a PolynomialLeading Coefficientx-Intercepts and MultiplicityEnd Behavior of Polynomials
Degree of a Polynomial
The degree of a polynomial is one of its most important characteristics. It tells us the highest power of the variable present in the polynomial. For example, in the polynomial \(-3x^3 - 6x^2 + 12x\), the highest power of the variable \(x\) is \(x^3\). This makes the degree of this polynomial 3. This is discovered by looking at the expanded form of the polynomial, which in our case is from the term \(-3x^3\).
Understanding the degree helps us make numerous predictions about the polynomial function. It sets the stage for analyzing the end behavior of the polynomial and also indicates how many roots, or solutions, the polynomial might have, which often equals the polynomial's degree (though not always in the real sense). In our example, the degree being 3 suggests that there are potentially three real roots or intercepts.
Understanding the degree helps us make numerous predictions about the polynomial function. It sets the stage for analyzing the end behavior of the polynomial and also indicates how many roots, or solutions, the polynomial might have, which often equals the polynomial's degree (though not always in the real sense). In our example, the degree being 3 suggests that there are potentially three real roots or intercepts.
Leading Coefficient
The leading coefficient is the coefficient of the term with the highest power in a polynomial. It's vital because it influences the polynomial's growth and the direction of the end behavior of its graph. In \(-3x^3 - 6x^2 + 12x\), the leading coefficient is \(-3\) associated with the term \(-3x^3\).
This negative leading coefficient indicates that the polynomial will reflect over the x-axis compared to a positive-leading polynomial with the same degree. It will also help identify how sharply the curve will ascend or descend as \(x\) extends towards positive or negative infinity.
This negative leading coefficient indicates that the polynomial will reflect over the x-axis compared to a positive-leading polynomial with the same degree. It will also help identify how sharply the curve will ascend or descend as \(x\) extends towards positive or negative infinity.
x-Intercepts and Multiplicity
Finding the \(x\)-intercepts is a crucial step when graphing polynomials. These are the values of \(x\) where the polynomial equals zero. For the polynomial \(n(x) = -3x(x+2)(x-4)\), setting \(n(x) = 0\) leads us to solving \(-3x(x+2)(x-4) = 0\).
- For \(x = 0\), the expression becomes \(-3 imes 0 = 0\).
- For \(x+2=0\), solving gives \(x = -2\).
- For \(x-4=0\), solving gives \(x = 4\).
End Behavior of Polynomials
End behavior describes how the values of a polynomial function behave as \(x\) approaches positive or negative infinity. It is primarily dictated by the degree and the leading coefficient of the polynomial.
In our example with degree 3 and leading coefficient \(-3\), which is odd and negative respectively, as \(x\) goes to \(+\infty\), \(n(x)\) approaches \(-\infty\). As \(x\) goes to \(-\infty\), \(n(x)\) approaches \(+\infty\). This tells us:
In our example with degree 3 and leading coefficient \(-3\), which is odd and negative respectively, as \(x\) goes to \(+\infty\), \(n(x)\) approaches \(-\infty\). As \(x\) goes to \(-\infty\), \(n(x)\) approaches \(+\infty\). This tells us:
- The left end of the graph will go upwards.
- The right end of the graph will go downwards.
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