Problem 62 - College Algebra

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. Th \(y\) -intercept is (0,9) . The \(x\) -intercepts are (-3,0) , (3,0) . 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 \(f(x) = -x^2 + 9\).

Step 1: Identify Key Information

We are given that the polynomial function has a degree of 2, which implies it is a quadratic polynomial of the form \(f(x) = ax^2 + bx + c\). The leading coefficient is either \(1\) or \(-1\). The \(x\)-intercepts are \(-3\) and \(3\), and the \(y\)-intercept is \(9\).

Step 2: Use X-Intercepts to Form Polynomial

The \(x\)-intercepts \((-3, 0)\) and \((3, 0)\) mean the factors of the polynomial are \((x + 3)\) and \((x - 3)\). Therefore, the polynomial can be written as \(f(x) = a(x + 3)(x - 3)\).

Step 3: Expand the Polynomial Expression

Expand the factors to get a standard quadratic form: \(f(x) = a(x^2 - 9)\). Therefore, \(f(x) = ax^2 - 9a\).

Step 4: Determine Leading Coefficient Using Y-Intercept

Given the \(y\)-intercept \((0, 9)\), substitute \(x = 0\) and \(f(x) = 9\) into the polynomial. Therefore, \(9 = a(0^2 - 9) = -9a\). Solving for \(a\) gives \(a = -1\).

Step 5: Analyze End Behavior and Verify Correctness

With \(a = -1\), the polynomial functions as \(f(x) = -x^2 + 9\). The end behavior \(f(x) \rightarrow -\infty\) as \(x \rightarrow \pm\infty\) matches the description of a downward-opening parabola, confirming \(a = -1\) is correct.

Key Concepts

Quadratic Polynomial

A quadratic polynomial is a type of polynomial that involves a degree of 2. This means the highest power of the variable, typically denoted as \( x \), is squared. The general form of a quadratic polynomial is \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). The coefficient \( a \) is particularly important because it affects the direction and width of the parabola that the quadratic function graphically represents.
In the exercise provided, the polynomial has a degree of 2, which confirms it is a quadratic polynomial. Knowing this helps us anticipate the shape of its graph, which should be a parabola. Additionally, the sign of \( a \) impacts whether the parabola opens upwards or downwards. If \( a \) is positive, it opens upwards; if \( a \) is negative, it opens downwards. This insight is essential for understanding and predicting the behavior of quadratic polynomials when analyzing their graphs or solving related problems.

Graphing Polynomial Functions

Graphing polynomial functions involves plotting a curve based on the function's equation derived from its polynomial expression. For quadratic polynomials, graphing yields a parabolic curve. Different characteristics of the polynomial, such as its coefficients, help determine the exact nature of this parabola.

• **Vertex:** The point where the parabola changes direction. In vertex form, this is easily identifiable.
• **Axis of Symmetry:** A vertical line that divides the parabola into two mirror-image halves, located at \( x = -\frac{b}{2a} \).
• **Intercepts:** Points where the graph intersects the axes, providing critical information about the behavior of the polynomial.

The given quadratic equation from the exercise was \( f(x) = -x^2 + 9 \). Notably, the leading coefficient is \(-1\), indicating that the parabola opens downwards. Knowing how to convert the function into a graph involves identifying intercepts, assessing symmetry, and determining the parabola's direction; all critical steps in building a complete graphical representation of the polynomial function.

Intercepts

Intercepts are critical points on a graph where the function intersects the axes. Quadratic polynomials, such as the one described in the exercise, have both x-intercepts and a y-intercept.
**X-Intercepts:** These occur where the graph crosses the x-axis. For each x-intercept \( (a, 0) \), the y-value is zero. To find x-intercepts, one can factorize the quadratic polynomial, as factors reveal roots of the function. In our exercise example, the x-intercepts occur at \((-3,0)\) and \((3,0)\), as derived from the factors \((x + 3)(x - 3)\). These points indicate where the parabola crosses the x-axis.
**Y-Intercept:** This occurs where the graph intersects the y-axis. At this point, the x-value is zero. Substituting \( x = 0 \) in the polynomial equation \( f(x) \) determines the value of the y-intercept. The exercise provides this point as \((0,9)\), implying the parabola cuts the y-axis at \( y = 9 \).
Understanding intercepts is crucial for accurately plotting polynomial functions and analyzing their characteristics. They provide specific anchor points on the graph, enabling a clearer understanding of the polynomial's behavior across its domain.