Problem 61 - College Algebra with Corequisite Support

Question

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or -1 . There may be more than one correct answer. The \(y\) - intercept is (0,-4) . The \(x\) - intercepts are \((-2,0),(2,0) .\) Degree is 2 End behavior: as \(x \rightarrow-\infty, f(x) \rightarrow \infty ;\) as \(x \rightarrow \infty\), \(f(x) \rightarrow \infty\)

Step-by-Step Solution

Answer

The polynomial function is y = x^2 - 4.

Step 1: Identify the Roots

Given the x-intercepts (-2, 0) and (2, 0), these indicate the roots of the polynomial function are x = -2 and x = 2. This suggests factors of (x + 2) and (x - 2).

Step 2: Write the Polynomial Expression

Use the identified roots to write the polynomial as y = a(x + 2)(x - 2).

Step 3: Determine Leading Coefficient (a)

Since the degree is 2 (quadratic) and as x approaches both positive and negative infinity, the function approaches positive infinity, the leading coefficient must be positive. Hence, a must be 1.

Step 4: Expand the Polynomial Expression

Using the factors (x + 2) and (x - 2), expand the expression y = 1(x + 2)(x - 2). This can be expanded to y = (x^2 - 4).

Step 5: Verify the y-intercept

Check if the y-intercept at (0, -4) is satisfied. Substitute x = 0 in y = x^2 - 4, giving y = (0)^2 - 4 = -4, which matches the y-intercept.

Key Concepts

Graph Intercepts

When analyzing polynomial functions, understanding graph intercepts is crucial. There are two types of intercepts to consider:

• Y-Intercept: This is where the graph crosses the y-axis. Mathematically, it's the point where the x-value is zero. In our example, the y-intercept is at (0, -4), which tells us that when we substitute 0 for x in the polynomial, the result should be -4.
• X-Intercepts: These are points where the graph crosses the x-axis. Here, the y-value is zero. For this exercise, the x-intercepts are (-2, 0) and (2, 0). These points indicate the roots of the polynomial. They show where the polynomial equals zero, revealing the factors of the polynomial as (x + 2) and (x - 2).

These intercepts are key to determining the structure of the polynomial function and are instrumental in graph sketching.

Leading Coefficient

The leading coefficient plays a significant role in shaping a polynomial function. It is the number that multiplies the highest degree term in the polynomial. In our problem, the leading coefficient can only be 1 or -1.

Specifically for this function:

• The degree is identified as 2, making it a quadratic function.
• Given that the function approaches positive infinity as x approaches both positive and negative infinity, a positive leading coefficient is confirmed.

This restricts the leading coefficient \(a\) to 1, as a positive quadratic function opens upwards. Choosing the correct leading coefficient ensures the proper end behavior of the polynomial.

Degree of Polynomial

The degree of a polynomial is determined by its highest exponent. It shows the highest power to which the variable is raised. In this exercise, the degree is given as 2, which characterizes the function as quadratic. Quadratic functions have several notable traits:

• A U-shaped graph, otherwise known as a parabola.
• Exactly two roots or x-intercepts, aligning with our given intercepts (-2,0) and (2,0).

The degree also informs us about the highest number of turns the graph can make and the number of x-intercepts. With a degree of 2, the quadratic function will typically have a single turning point.

End Behavior

Understanding the end behavior of a polynomial function tells us how the function behaves as x approaches positive or negative infinity. For the quadratic polynomial in this example, where the degree is even (2), and the leading coefficient is positive, the end behavior reflects a very specific pattern:

• As \(x ightarrow -\infty\), the function \(f(x) ightarrow \infty\).
• As \(x ightarrow \infty\), the function \(f(x) ightarrow \infty\).

This means the graph opens upwards, indicating that both the left and right ends of the graph rise to infinity. Recognizing this pattern helps to predict the direction in which the arms of the graph move and is crucial for sketching or understanding the shape of polynomial graphs.