Problem 99 - Precalculus : Building Concepts and Connections

Question

Can you write down an expression for a quadratic function whose \(x\) -intercepts are given by (2,0) and (3,0)\(?\) Is there more than one possible answer? Explain.

Step-by-Step Solution

Answer

Yes, there are infinitely many quadratic functions due to the variable 'a', with the same roots at (2,0) and (3,0). The general form is \(f(x) = a(x - 2)(x - 3)\), where 'a' can be any real number.

Step 1: Identify the Roots

The x-intercepts given are (2,0) and (3,0). This means the roots of the function are 2 and 3.

Step 2: Formulate the quadratic function from the roots

When the roots of a quadratic function \(f(x) = a(x - h)^2 + k\) are known, the function can be formulated. Here, a change in the function's form to \(f(x) = a(x - r_1)(x - r_2)\) where \(r_1\) and \(r_2\) are the roots of the function is helpful. Plugging in the roots yields \(f(x) = a(x - 2)(x - 3)\)

Step 3: Determine the coefficient

There is indeed more than one answer to this question since the coefficient 'a' in the quadratic equation can take any value and still maintain the same roots. It would simply stretch or shrink the graph, or flip it if negative. In other words, the coefficient 'a' alters the steepness or direction of the parabola but not the roots. Thus, a family of quadratic functions can be developed by taking different values for 'a'.

Key Concepts

Understanding x-intercepts

X-intercepts are the points where a graph crosses or touches the x-axis. For a quadratic function, which typically forms a parabola when graphed, these points are crucial as they show where the output value (y) is zero. In mathematical terms, x-intercepts are solutions to the equation when the function is set to zero.

To find the x-intercepts of a quadratic function, you must solve for x when the function equals zero. This often involves factoring the quadratic expression if possible.

• Example: Given roots (2,0) and (3,0), the equation can be written as \((x-2)(x-3)=0\).
• Solving gives two solutions or x-intercepts at x = 2 and x = 3.

These intercepts help in visualizing the graph of the function, giving you points where the parabola intersects the x-axis, offering a clear representation of real-number solutions.

Roots of a quadratic equation

The roots of a quadratic equation are the values of x for which the quadratic expression equals zero. These roots are equivalent to the x-intercepts of the quadratic function. More formally, if you have a quadratic function \(f(x) = ax^2 + bx + c\), the roots are the solutions to the equation \(ax^2 + bx + c = 0\).

Finding the roots can be achieved through various methods such as:

• Factoring, when the quadratic can be expressed as a product of binomials.
• Using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Completing the square, a method particularly useful when the quadratic is difficult to factor.

Each method has specific scenarios where it's most effective. For instance, in our exercise, the roots are clearly given as simple real numbers, allowing you to directly factor the quadratic equation, as seen when we expressed it in the form \((x - r_1)(x - r_2)\). This particular approach not only helps identify roots but also construct the quadratic function from known x-intercepts.

Important parabola properties

Parabolas are U-shaped curves that are symmetrical and can open upwards or downwards depending on the leading coefficient (a) in the quadratic equation \(f(x) = ax^2 + bx + c\).

Some essential properties of parabolas include:

• **Vertex**: The peak point of the parabola, where it changes direction. It's also the minimum or maximum point of the function, depending on whether the parabola opens upward or downward.
• **Axis of symmetry**: A vertical line that passes through the vertex, dividing the parabola into two mirror images. This line can be identified by the formula \(x = -\frac{b}{2a}\).
• **Direction**: Determined by the sign of the leading coefficient 'a'. If 'a' is positive, the parabola opens upwards; if negative, it opens downwards.

The stretches or shrinks in the parabola are controlled by the absolute value of 'a'. A larger \(|a|\) means a steeper graph, while a smaller \(|a|\) results in a wider graph. Despite these changes, the x-intercepts remain constant since they depend solely on the roots. Understanding these properties is crucial when analyzing or graphing quadratic functions.