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Chapter 3: Problem 65

Can the graph of a polynomial function have no \(y\) -intercept? Explain,

Short Answer

Expert verified
No, the graph of a polynomial function always has a y-intercept. This is because at \(x = 0\), the equation simplifies to the constant term of the polynomial, which gives the y-intercept of the graph.

Step by step solution

01

Understanding the y-intercept in a function

The y-intercept of a function is the point at which the graph of the function intersects or touches the y-axis. In mathematical terms, the y-intercept of a function is the value of the function when \(x = 0\). That is to say, the y-intercept of a function \(f(x)\) is found by determining the value of \(f(0)\). Therefore, a function has a y-intercept if it is defined at \(x = 0\).
02

Understanding Polynomial Functions

A polynomial function is a function that can be expressed in the form \(f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0\), where \(n\) is a non-negative integer and \(a_0, a_1, ..., a_n\) are coefficients. For every polynomial function, if we substitute \(x = 0\), we get \(f(x) = a_0\), which is a constant.
03

Answering the Exercise

From step 2, we know substituting \(x = 0\) into a polynomial function, the result is \(a_0\), which is the constant term in the polynomial. This means for all polynomial functions, the graph crosses or touches the y-axis at \(y = a_0\). Therefore, a polynomial function always has a y-intercept.

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Key Concepts

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

Polynomial Function
A polynomial function is essentially a mathematical expression that consists of variables鈥攃alled indeterminates鈥攁nd coefficients. The general form of a polynomial function is represented as follows: \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_2x^2 + a_1x + a_0 \). Here:
  • \( n \) is a non-negative integer.
  • \( a_n, a_{n-1}, ..., a_0 \) are coefficients, where \( a_n eq 0 \) if \( n \) is the degree of the polynomial.
  • \( a_0 \) is the constant term of the polynomial.
Polynomial functions are very flexible and are used in all sorts of problems, ranging from simple to very complex. What makes them unique is that they can take on different shapes based on the degree and the coefficients of the polynomial.
Y-Axis Intersection
The y-axis intersection, commonly referred to as the y-intercept, is a significant concept when studying functions. This point on a graph represents where the function crosses the y-axis. It gives us a specific value of the function when the input, \( x \), is zero.Visualizing the graph of a function, the y-intercept is the spot where the line or curve touches the vertical axis. Practically speaking, this intersection point is extremely useful as it gives us a starting point when graphing a polynomial, allowing for easier visualization.
X = 0 Substitution
The concept of substituting \( x = 0 \) into a function stems from the need to find the y-intercept, a crucial part of graph analysis. When analyzing polynomial functions, plugging in \( x = 0 \) allows us to isolate the constant term, \( a_0 \).By performing this substitution:
  • The term \( x^n \) becomes \( 0^n = 0 \), reducing all terms with \( x \) as a factor to zero.
  • Thus, \( f(0) = a_0 \), which is simply the constant term in the polynomial.
This process shows why polynomial functions always have a y-intercept: because \( f(0) = a_0 \), which is a constant and always defined.
Constant Term in Polynomial
The constant term in a polynomial, often denoted as \( a_0 \), holds special importance. This is because it directly represents the y-intercept of the polynomial function. Unlike other terms in a polynomial, the constant term is not tied to the variable \( x \). As a result:
  • When \( x = 0 \), all other terms vanish, leaving only the constant term.
  • It defines the intersection of the graph with the y-axis, denoted as \( y = a_0 \).
In practical terms, knowing \( a_0 \) instantly gives you the y-intercept without more complex calculations. This highlights why any polynomial function will always have a y-intercept.

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Most popular questions from this chapter

In Exercises \(27-34,\) find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=x^{3}+7 x^{2}-4 x-28$$

Use a graphing utility to obtain a complete graph for each polynomial function in Exercises \(58-61 .\) Then determine the number of real zeros and the number of nonreal complex zeros for each function. $$ f(x)=x^{6}-64 $$

A herd of 100 elk is introduced to a small island. The number of elk, \(N(t),\) after \(t\) years is described by the polynomial function \(N(t)=-t^{4}+21 t^{2}+100\) a. Use the Leading Coefficient Test to determine the graphs end behavior to the right. What does this mean about what will eventually happen to the elk population? b. Graph the function. c. Graph only the portion of the function that serves as a realistic model for the elk population over time. When does the population become extinct?

Begin by deciding on a product that interests the group because you are now in charge of advertising this product. Members were told that the demand for the product varies directly as the amount spent on advertising and inversely as the price of the product. However, as more money is spent on advertising, the price of your product rises. Under what conditions would members recommend an increased expense in advertising? Once you've determined what your product is, write formulas for the given conditions and experiment with hypothetical numbers. What other factors might you take into consideration in terms of your recommendation? How do these factor affect the demand for your product?

In Exercises \(55-56,\) use a graphing utility to determine upper and lower bounds for the zeros of \(f .\) Does synthetic division verify your observations? $$ f(x)=2 x^{4}-7 x^{3}-5 x^{2}+28 x-12 $$
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