Problem 31 - Precalculus : Building Concepts and Connections

Question

Find a possible expression for a quadratic function \(f(x)\) having the given zeros. There can be more than one correct answer. $$x=\frac{1}{2} \text { and } x=3$$

Step-by-Step Solution

Answer

The equation of the quadratic function with roots \(x = \frac{1}{2}\) and \(x = 3\) can be \(f(x) = x^2 - \frac{7}{2}x + \frac{3}{2}\).

Step 1: Identify the roots

Our given roots are \(x = \frac{1}{2}\) and \(x = 3\).

Step 2: Plug the roots into the quadratic formula

To find our quadratic function, we will simply substitute the roots h and k in the formula \(f(x) = a(x-h)(x-k)\). Let's use a=1 for simplicity. Substituting h as \(\frac{1}{2}\) and k as 3 into the formula, we have \(f(x) = (x - \frac{1}{2})(x - 3)\).

Step 3: Simplify the expression

Multiplying out the expression we get \(f(x) = x^2 - \frac{7}{2}x + \frac{3}{2}\).

Key Concepts

Quadratic Formula

Understanding the quadratic formula is essential for solving quadratic equations and finding the zeros of a quadratic function. The quadratic formula is expressed as:

\[\begin{equation}x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\end{equation}\]
This formula is derived from the standard form of a quadratic equation, which is:\[\begin{equation}ax^2 + bx + c = 0\end{equation}\]
Given a quadratic equation in standard form, you can use the quadratic formula to find the values of x that make the equation true—these are called the roots or zeros. The terms under the square root, known as the discriminant \[\begin{equation}D = b^2 - 4ac\end{equation}\]
, can tell us about the nature of the roots. A positive discriminant indicates two distinct real roots, a discriminant of zero indicates one real root (also called a repeated or double root), and a negative discriminant indicates two complex roots.

Roots of a Quadratic Equation

When we talk about the roots of a quadratic equation, we’re talking about the values of x that satisfy the equation \[\begin{equation}ax^2 + bx + c = 0 \end{equation}\]
In our example, the roots are \[\begin{equation}x = \frac{1}{2} \text{ and } x = 3\end{equation}\]
These roots are the 'x-intercepts' or 'zeros' of a quadratic function because they are the points where the graph of the function crosses the x-axis. Finding these is crucial for graphing the quadratic function, and they provide important information about the function's characteristics.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a fundamental skill in algebra. It involves combining like terms and using distributive properties to make expressions easier to work with. In our solution, we simplified the expression \[\begin{equation}(x - \frac{1}{2})(x - 3)\end{equation}\]
by expanding the multiplication and combining like terms to get:\[\begin{equation}f(x) = x^2 - \frac{7}{2}x + \frac{3}{2}\end{equation}\]
This simplification step is critical for interpreting the standard form of quadratic functions and for further calculations like evaluating the function for given values of x or finding the vertex form.

Finding a Quadratic Function

A quadratic function can be determined when the zeros are given, as these zeros are the roots or solutions of the quadratic equation. By knowing that a quadratic function in its factored form looks like \[\begin{equation}f(x) = a(x - h)(x - k)\end{equation}\],
where \[\begin{equation}h \end{equation}\]
and\[\begin{equation}k\end{equation}\]
are the zeros, we can use this form to write the function. We simply replace h and k with the given zeros and choose a suitable value for 'a' (often taken as 1 for simplicity). We then expand this to find the quadratic function in standard form. This conversion from factored form to standard form helps us understand the behavior of the function and is useful for graphing and analyzing the function's properties such as its vertex, axis of symmetry, and the direction it opens.