Problem 62 - College Algebra

Question

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 \((0,9) .\) 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 function is \( -x^2 + 9 \).

Step 1: Identify the Form of the Polynomial

Since the degree of the polynomial is 2, the function is a quadratic and can be expressed generally as \( f(x) = ax^2 + bx + c \). The end behavior indicates that as \( x \to \pm\infty \), \( f(x) \to -\infty \), suggesting the leading coefficient, \( a \), is negative.

Step 2: Determine the Roots

The \(x\)-intercepts given are \((-3, 0)\) and \((3,0)\). This means the function can be expressed in factored form as \( f(x) = a(x + 3)(x - 3) \).

Step 3: Use the End Behavior

The end behavior indicates that the graph of the function opens downwards, confirming \( a = -1 \). Thus, the polynomial now is \( f(x) = -(x + 3)(x - 3) \).

Step 4: Expand the Polynomial

Expanding the factored polynomial: \( f(x) = -(x^2 - 9) = -x^2 + 9 \).

Step 5: Check the Y-Intercept

The \(y\)-intercept is given as \((0,9)\). When \(x = 0\), \( f(0) = -0^2 + 9 = 9 \), which matches the given \(y\)-intercept.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function that is represented by the equation \( f(x) = ax^2 + bx + c \).
It always has a degree of 2.
This means the highest exponent of the variable, in this case \( x \), is 2. Here are some important characteristics of quadratic functions:

• They can open upwards or downwards depending on the sign of the leading coefficient \( a \).
• If \( a \) is positive, the parabola opens upwards.
• If \( a \) is negative, the parabola opens downwards.
• The graph is a parabola, which is a symmetrical curve.

Understanding the quadratic structure helps us to predict the general shape and direction of the graph. In our exercise, since the end behavior directs downwards, the function demonstrates a negative leading coefficient, making the structure \( f(x) = -x^2 + bx + c \).
This tells us the graph will be a downward opening parabola.

X-Intercepts

X-intercepts are the points where the graph of a function crosses the x-axis.
In other words, they are the solutions of the equation \( f(x) = 0 \).
These points are also known as roots or zeroes of the function.

• For our quadratic function \( f(x) \), the given x-intercepts are \((-3, 0)\) and \((3, 0)\).
• This indicates the factors of the equation are \((x + 3)\) and \((x - 3)\).
  

To find these intercepts in any quadratic function, you can set the quadratic equation \( ax^2 + bx + c = 0 \) and solve for \( x \).
In factored form, this quadratic corresponds to \( f(x) = a(x + 3)(x - 3) \).
These intercepts help in shaping the function's graph and offer crucial information about where the graph will cross the x-axis.

Y-Intercept

The \( y \)-intercept of a function is where the graph crosses the \( y \)-axis. It occurs when \( x = 0 \).
This means you simply need to evaluate the function at \( x = 0 \) to determine the \( y \)-intercept.
For the quadratic function \( f(x) = ax^2 + bx + c \), the \( y \)-intercept is the constant term \( c \).

• In our exercise, the given \( y \)-intercept is \((0, 9)\).
• This indicates that when substituting \( x = 0 \) into the polynomial \( f(x) = -(x + 3)(x - 3) \), we should get 9.
  
• Calculating, \( f(0) = -0^2 + 9 = 9 \), confirming our \( y \)-intercept is correct.

The \( y \)-intercept is crucial because it gives a starting point for graphing the function and verifying the formula is correct based on the known points.

End Behavior

End behavior describes how the values of a function change as \( x \) moves towards positive or negative infinity.
For quadratic functions, this behavior helps determine how the parabola opens and in which direction it extends.

• In our example, the end behavior was described as: as \( x \to -\infty \), \( f(x) \to -\infty \) and as \( x \to \infty \), \( f(x) \to -\infty \).
• These observations confirm that our function is a downward-opening parabola.
• This type of end behavior typically happens when the leading coefficient \( a \) is negative.

Understanding end behavior allows you to predict the general direction and orientation of any quadratic graph without plotting many points.
It also confirms the suitability of the derived quadratic equation for the given polynomial characteristics.