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Chapter 8: Problem 8

Describe how \(g(x)\) and \(h(x)\) relate to \(f(x)\) $$ \begin{array}{l} f(x)=x^{5}-3 x^{2}+4 \\ g(x)=-x^{5}+3 x^{2}-4 \\ h(x)=x^{5}-3 x^{2}-2 \end{array} $$

Short Answer

Expert verified
g(x)=-f(x) and h(x)=f(x)-6.

Step by step solution

01

Identify the given functions

Examine the functions given in the problem: \[f(x) = x^5 - 3x^2 + 4\] \[g(x) = -x^5 + 3x^2 - 4\] \[h(x) = x^5 - 3x^2 - 2\]
02

Compare coefficients and constants of g(x) with f(x)

Compare each term in \(g(x)\) with \(f(x)\): \(f(x) = x^5 - 3x^2 + 4\) has a coefficient of 1 for \(x^5\), -3 for \(x^2\) and a constant of 4. \(g(x)\) has coefficients: -1 for \(x^5\), +3 for \(x^2\), and a constant of -4. Note that \(g(x)\) is the opposite of \(f(x)\), so we can write: \[g(x) = -f(x)\]
03

Compare coefficients and constants of h(x) with f(x)

Compare each term in \(h(x)\) with \(f(x)\): \(f(x) = x^5 - 3x^2 + 4\) has a coefficient of 1 for \(x^5\), -3 for \(x^2\) and a constant of 4. \(h(x)\) has the same coefficients for \(x^5\) and \(x^2\) terms but a different constant of -2. This shows that \(h(x)\) is just \(f(x)\) shifted downward by 6 units since \[4 - (-2) = 6\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Functions
Polynomial functions are expressions composed of variables and coefficients, combined using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. In the given exercise, the function \(f(x) = x^5 - 3x^2 + 4\) is a polynomial of degree 5, meaning the highest power of \(x\) is 5. Polynomial functions can represent a variety of shapes and behaviors, depending on their degree and the values of their coefficients and constants. To identify polynomial functions, look for terms of the form \(a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\), where \(a_n\) are the coefficients and \(x\) are the variable's powers.
Function Transformation
Function transformations involve shifting and changing the shape of a graph of a function. In the provided exercise, the relationship between \(f(x), g(x),\) and \(h(x)\) can be understood through transformations. Specifically:
  • \(g(x) = -f(x)\): This represents a reflection of \(f(x)\) across the x-axis. Each point on the graph of \(f(x)\) is mirrored. For example, if \(f(a) = b\), then \(g(a) = -b\).
  • \(h(x)\) changes only the constant term compared to \(f(x)\): \(h(x) = f(x) - 6\). This indicates a vertical shift. Since the constant in \(f(x)\) is 4 and in \(h(x)\) it is -2, \(f(x)\) is shifted down by \(4 - (-2) = 6\) units.
Understanding these transformations helps in graphing functions and predicting their behavior without detailed calculations.
Coefficients and Constants
Coefficients and constants are vital parts of polynomial functions. Coefficients are the numbers multiplying the variables, and constants are the terms without variables. Here's their role in the given functions:
  • In \(f(x) = x^5 - 3x^2 + 4\), the coefficients are 1 for \(x^5\) and -3 for \(x^2\), and the constant is 4. These coefficients determine the function's shape and direction.
  • In \(g(x) = -x^5 + 3x^2 - 4\), the coefficients are -1 for \(x^5\) and +3 for \(x^2\), and the constant is -4. The negation of each term's coefficient flips the function vertically.
  • In \(h(x) = x^5 - 3x^2 - 2\), the coefficients match those of \(f(x)\) (1 for \(x^5\) and -3 for \(x^2\)), but the constant term is different (-2). The difference in constants results in a vertical shift of the function's graph.
Recognizing the role of coefficients and constants is crucial in analyzing and manipulating polynomial functions. Changes in coefficients affect the rate of growth or decay of the function, while constants shift the function up or down on the graph.

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Most popular questions from this chapter

Put each of the following quadratics into standard form. a. \(f(x)=(x+3)(x-1)\) b. \(P(t)=(t-5)(t+2)\) c. \(H(z)=(2+z)(1-z)\)

Transform the function \(f(x)=3 x^{2}\) into a new function \(h(x)\) by shifting \(f(x)\) horizontally to the right four units, reflecting the result across the \(x\) -axis, and then shifting it up by five units. a. What is the equation for \(h(x) ?\) b. What is the vertex of \(h(x) ?\) c. What is the vertical intercept of \(h(x) ?\)

A pilot has crashed in the Sahara Desert. She still has her maps and knows her position, but her radio is destroyed. Her only hope for rescue is to hike out to a highway that passes near her position. She needs to determine the closest point on the highway and how far away it is. a. The highway is a straight line passing through a point 15 miles due north of her and another point 20 miles due east. Draw a sketch of the situation on graph paper, placing the pilot at the origin and labeling the two points on the highway. b. Construct an equation that represents the highway (using \(x\) for miles east and \(y\) for miles north). c. Now use the Pythagorean Theorem to describe the square of the distance, \(d,\) of the pilot to any point \((x, y)\) on the highway. d. Substitute the expression for \(y\) from part (b) into the equation from part (c) in order to write \(d^{2}\) as a quadratic in \(x\) e. If we minimize \(d^{2}\), we minimize the distance \(d\). So let \(D=d^{2}\) and write \(D\) as a quadratic function in \(x\). Now find the minimum value for \(D\). f. What are the coordinates of the closest point on the highway, and what is the distance, \(d\), to that point?

In each part, construct a polynomial function with the indicated characteristics. a. Crosses the \(x\) -axis at least three times b. Crosses the \(x\) -axis at \(-1,3,\) and 10 c. Has a \(y\) -intercept of 4 and degree of 3 d. Has a \(y\) -intercept of -4 and degree of 5

Complete the following table, and then summarize your findings. $$ \begin{array}{ll} i^{1}=\sqrt{-1}=i & i^{5}=? \\ i^{2}=i \cdot i=-1 & i^{6}=? \\ i^{3}=i \cdot i^{2}=? & i^{7}=? \\ i^{4}=i^{2} \cdot i^{2}=? & i^{8}=? \end{array} $$
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