Problem 65 - College Algebra

Question

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

Step-by-Step Solution

Answer

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 1 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 \(x = 0\).

Step 2 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.

Step 3 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.

Key Concepts

Polynomial Function

A polynomial function is essentially a mathematical expression that consists of variables—called indeterminates—and 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.