Problem 109
Question
Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. What is the relationship between the two graphs? Use the Binomial Theorem to write the polynomial function \(g\) in standard form. \(f(x)=-x^{3}+3 x^{2}-4\) \(g(x)=f(x+5)\)
Step-by-Step Solution
Verified Answer
The graph of \(g\) is a shift of the graph of \(f\) five units to the left. The polynomial function \(g\) in standard form is \( -x^{3}-15x^{2}-75x-124\).
1Step 1 - Graph the functions
First, use a graphing utility to graph \(f(x)=-x^{3}+3 x^{2}-4\) and \(g(x)=f(x+5)\) on the same axis to visually ascertain the relationship between the two.
2Step 2 - Assess the visual relationship between the graphs
Once the functions have been graphed, compare the two. Because \(g(x)\) is essentially \(f(x+5)\), its graph will appear as a horizontal shift of the graph of \(f(x)\) 5 units to the left.
3Step 3 - Use the Binomial Theorem to write \(g\) in standard form
The most challenging part comes in simplifying \(g(x) = f(x+5)\), also written as \(- (x+5)^{3}+3(x+5)^{2}-4\). This is where we will apply the Binomial Theorem to break down and simplify the polynomial.
4Step 4 - Simplify the terms
Applying the Binomial Theorem and simplification result in \( -x^{3}-15x^{2}-75x-124\)
Key Concepts
Binomial TheoremStandard Form of a PolynomialHorizontal Shift
Binomial Theorem
When studying polynomial functions, the Binomial Theorem is a crucial mathematical tool that allows us to expand expressions raised to a power. Put simply, it turns something like \( (a+b)^n \) into a sum involving terms of the form \( a^k b^{n-k} \) where \( k \) is a whole number between 0 and \( n \), inclusive. This is particularly useful when dealing with polynomial functions where a binomial expression is raised to a higher power.
For example, let's consider the function \( g(x)=f(x+5) \) from the given problem. Here, we need to expand \( (x+5)^3 \) and \( (x+5)^2 \) using the Binomial Theorem. The theorem states that \( (x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \) where \( \binom{n}{k} \) represents the binomial coefficients or 'choose' function, found on Pascal's triangle or calculated using factorials. This systematic approach allows us to simplify polynomial expressions like \( g(x) \) into a form that is much more manageable for further analysis and graphing.
For example, let's consider the function \( g(x)=f(x+5) \) from the given problem. Here, we need to expand \( (x+5)^3 \) and \( (x+5)^2 \) using the Binomial Theorem. The theorem states that \( (x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \) where \( \binom{n}{k} \) represents the binomial coefficients or 'choose' function, found on Pascal's triangle or calculated using factorials. This systematic approach allows us to simplify polynomial expressions like \( g(x) \) into a form that is much more manageable for further analysis and graphing.
Standard Form of a Polynomial
Polynomials are algebraic expressions that consist of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The standard form of a polynomial organizes the terms in descending order of their exponents on the variable.
For example, a cubic polynomial in the standard form will look like \( ax^3+bx^2+cx+d \) where \( a, b, c \) and \( d \) are coefficients and \( a \) is nonzero. The significance of writing a polynomial in standard form is that it reveals the highest degree of the polynomial (the leading term) and helps us anticipate certain properties of its graph, such as the possible number of turns. Applying this to the function \( g(x) \) from our example, once expanded using the Binomial Theorem, would result in the polynomial being organized in the standard form: \( -x^3-15x^2-75x-125 \) showing the coefficients clearly for each degree of \( x \) and facilitating a more straightforward analysis of its characteristics and graph.
For example, a cubic polynomial in the standard form will look like \( ax^3+bx^2+cx+d \) where \( a, b, c \) and \( d \) are coefficients and \( a \) is nonzero. The significance of writing a polynomial in standard form is that it reveals the highest degree of the polynomial (the leading term) and helps us anticipate certain properties of its graph, such as the possible number of turns. Applying this to the function \( g(x) \) from our example, once expanded using the Binomial Theorem, would result in the polynomial being organized in the standard form: \( -x^3-15x^2-75x-125 \) showing the coefficients clearly for each degree of \( x \) and facilitating a more straightforward analysis of its characteristics and graph.
Horizontal Shift
The concept of horizontal shift is a transformation that occurs in the graphing of functions and it's a movement to the left or right along the x-axis. It is vital for understanding how modifying the input of a function affects its graph. In the case of a function \( f(x) \) and a transformation that adds a number \( h \) to the input, the new function \( g(x) = f(x+h) \) will represent a horizontal shift.
For positive values of \( h \) the graph of \( g \) will shift \( h \) units to the left, while for negative values, it will shift \( |h| \) units to the right. In the discussed exercise, \( g(x) = f(x+5) \) is a horizontal shift of the function \( f(x) \) by 5 units to the left. This is because we add 5 to the input, thus 'preponing' its effects on the graph. Recognizing horizontal shifts is particularly useful when comparing the graphs of the original function and its transformation, as it helps predict and understand the behavior without generating the full plot.
For positive values of \( h \) the graph of \( g \) will shift \( h \) units to the left, while for negative values, it will shift \( |h| \) units to the right. In the discussed exercise, \( g(x) = f(x+5) \) is a horizontal shift of the function \( f(x) \) by 5 units to the left. This is because we add 5 to the input, thus 'preponing' its effects on the graph. Recognizing horizontal shifts is particularly useful when comparing the graphs of the original function and its transformation, as it helps predict and understand the behavior without generating the full plot.
Other exercises in this chapter
Problem 108
Use sigma notation to write the sum. Then use a graphing utility to find the sum. $$1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdot \cdot \cdot-\frac{1}{128}$$
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Use sigma notation to write the sum. Then use a graphing utility to find the sum. $$\frac{1}{1^{2}}-\frac{1}{2^{2}}+\frac{1}{3^{2}}-\frac{1}{4^{2}}+\cdots \cdot
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