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Chapter 4: Problem 8

For the following exercises, identify whether the statement represents an exponential function. Explain. The height of a projectile at time \(t\) is represented by the function \(h(t)=-4.9 t^{2}+18 t+40\)

Short Answer

Expert verified
The function is not exponential; it's quadratic.

Step by step solution

01

Understanding Exponential Functions

An exponential function is typically in the form \( f(x) = a \, b^x \). Characteristics include a constant base \( b \) raised to a variable exponent \( x \).
02

Analyzing the Given Function

Look at the function \( h(t) = -4.9t^2 + 18t + 40 \). Identify the components: it involves \( t^2 \), which indicates a quadratic term, and additional terms with \( t \) and a constant.
03

Comparing with Exponential Characteristics

Observe that the given function is polynomial rather than exponential. It has a quadratic term \( t^2 \), indicating it is a quadratic function instead of an exponential one, which would require \( t \) in the exponent position.
04

Conclusion

The function \( h(t) = -4.9t^2 + 18t + 40 \) is not an exponential function but a quadratic function, as it is structured in the form \( at^2 + bt + c \).

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Key Concepts

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

Quadratic Functions
In mathematics, quadratic functions are a type of polynomial function that follow a specific pattern. They are commonly written in the standard form:
  • \( f(x) = ax^2 + bx + c \),
where \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). This ensures that the graph of the function is a parabola. Quadratic functions have a variety of important characteristics:
  • Vertex: The vertex is the highest or lowest point on the graph of the parabola. It is a crucial feature, as it represents the maximum or minimum value of the quadratic function.
  • Axis of Symmetry: This is a vertical line that passes through the vertex of the parabola, splitting it into two symmetrical halves. The equation of this line is often given by \( x = -\frac{b}{2a} \).
  • Roots or Zeroes: These are the values of \( x \) for which \( f(x) = 0 \). They can be found using the quadratic formula, factoring, or completing the square.
  • Direction of Opening: The parabola can open upwards or downwards depending on the sign of \( a \). If \( a > 0 \), the parabola opens upwards. If \( a < 0 \), it opens downwards.
Understanding these characteristics allows you to analyze and graph quadratic functions effectively.
Polynomial Functions
Polynomial functions encompass a wide variety of expressions that involve variables raised to non-negative integer exponents. A general polynomial function is expressed as:
  • \( P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \),
where \( n \) denotes the degree of the polynomial, and \( a_n \) are constant coefficients. Key features of polynomial functions include:
  • Degree and Leading Coefficient: The degree of a polynomial is the highest power of the variable \( x \), which indicates the polynomial's complexity. The leading coefficient is the coefficient of the term with the highest degree.
  • Behavior and End Behavior: The degree and leading coefficient of the polynomial determine the behavior of the graph, especially its end behavior (how the graph behaves as \( x \) approaches infinity or negative infinity).
  • Zeros: These are solutions to the equation \( P(x) = 0 \). The number of zeros is at most equal to the degree of the polynomial.
  • Smooth and Continuous: Polynomial functions are always smooth (no breaks in the graph) and continuous (the graph is unbroken) over their domain.
Quadratic functions are specifically second-degree polynomial functions, making them a subset of this broader category.
Characteristics of Functions
Functions in mathematics are equations that provide a unique output for every valid input. Understanding the characteristics of functions is essential for identifying different types. Some important characteristics include:
  • Domain and Range: The domain of a function consists of all possible input values, while the range includes all possible outputs. Understanding these helps in understanding the limitations of the function.
  • Intercepts: Intercepts are points where the graph of the function crosses the axes. The x-intercepts (or roots) are where the graph crosses the x-axis, and the y-intercept is where it crosses the y-axis.
  • Increasing and Decreasing Intervals: Functions can have segments where their values are increasing or decreasing. Identifying these intervals helps in understanding the overall behavior of the function.
  • Symmetry and Asymptotes: Some functions are symmetrical about an axis or a point, while others have asymptotes, which are lines that the graph approaches but never touches. Recognizing these characteristics helps in sketching and analyzing the graph of the function.
By identifying these characteristics, one is better capable of analyzing, interpreting, and applying different functions to real-world scenarios.

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Most popular questions from this chapter

For the following exercises, refer to Table 4.26. $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} & {6} \\\ \hline f(x) & {1125} & {1495} & {2310} & {3294} & {4650} & {6361} \\\ \hline\end{array}$$ Write the exponential function as an exponential equation with base \(e\).

For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table could represent a function that is linear, exponential, or logarithmic. $$\begin{array}{|c|c|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} & {9} & {10}\\\ \hline f(x) & {2} & {4.079} & {5.296} & {6.159} & {6.828} & {7.375} & {7.838} & { 8.238} & { 8.592} & { 8.908 }\\\ \hline\end{array}$$

Determine whether Table 4.32 could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points. $$\begin{array}{|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} \\ \hline f(x) & {3} & {0.9} & {0.27} & {0.081} \\ \hline\end{array}$$

For the following exercises, use the logistic growth model \(f(x)=\frac{150}{1+8 e^{-2 x}}\) Graph the model.

For the following exercises, refer to Table 4.29. $$\begin{array}{|c|c|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} \\ \hline f(x) & {7.5} & {6} & {5.2} & {4.3} & {3.9} & {3.1} & {2.9} \\ \hline\end{array}$$ Use the LOGarithm option of the REGression feature to find a logarithmic function of the form \(y=a+b \ln (x)\) that best fits the data in the table.
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