Problem 12 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For Problems 10 through 16, give the set of functions all having the speci ed characteristics. Example: the set of nonvertical lines passing through \((3,2) .\) Solution: \(y-2=m(x-3)\), so \(f(x)=m(x-3)+2\), where \(m\) is any constant. The set of parabolas with \(x\) -intercepts at \(x=0\) and \(x=3\)

Step-by-Step Solution

Answer

The set of parabolas with x-intercepts at \(x=0\) and \(x=3\) is given by the function \(f(x)=kx(x-3)\), where \(k\) is any constant.

Step 1: Determine Roots

The function will intercept the x-axis at \(x=0\) and \(x=3\). These are the roots of the function.

Step 2: Formulate the Function

The general form of a parabolic function whose roots are \(a\) and \(b\) is \(f(x)=k(x-a)(x-b)\), where \(k\) is any constant. So, replace \(a\) and \(b\) with \(0\) and \(3\) respectively.

Step 3: Simplify the Function

Here, \(a=0\) and \(b=3\). Thus, simplifying the function \(f(x)=k(x-0)(x-3)\) gives \(f(x)=kx(x-3)\).

Key Concepts

X-Intercepts of Parabolas

In the realm of quadratic functions, the concept of x-intercepts, also known as zeros of the function, is foundational. X-intercepts are the points at which a graph crosses the x-axis.

To find the x-intercepts of a parabola, we look for the values of x that make the quadratic function equal to zero. In the context of the given exercise, the parabola has been identified to cross the x-axis at x=0 and x=3. This information is crucial as it directly reveals the roots of the parabola.

Understanding the locations of the x-intercepts assists in sketching the parabola's shape and understanding its directional movement. If the quadratic coefficient is positive, the parabola opens upwards, and if negative, downwards.

Quadratic Functions

Quadratic functions represent one of the simplest forms of polynomial equations, characterized by the standard form f(x) = ax^2 + bx + c. These functions graph into parabolas, which are symmetric, U-shaped curves that either open upwards or downwards.

The leading coefficient a determines the parabola's direction; positive opens upwards, and negative opens downwards. The vertex is the highest or lowest point on a parabola, while the axis of symmetry is a vertical line that passes through the vertex and splits the parabola into two mirror images. For our exercise, the simpler factored form f(x) = k(x-a)(x-b) is used because the x-intercepts are given and serve as a faster way to write the quadratic function.

Roots of a Function

The roots or solutions of a function are the values of x for which the function equals zero. In quadratic functions, these roots are the same as the x-intercepts and can be found by setting the function equal to zero and solving for x.

The number of roots a quadratic function has can be determined by the discriminant b^2 - 4ac. If the discriminant is positive, there are two real roots; if zero, one real root; and if negative, no real roots. In the exercise, with specified roots of 0 and 3, we can see the function intersects the x-axis at two distinct points.

Solving Quadratic Equations

To solve quadratic equations, various methods are implemented depending on the form and complexity of the equation, including factoring, using the quadratic formula, completing the square, or graphing. Factoring is often the simplest method, as demonstrated in our exercise.

Given the roots, such as x=0 and x=3, creating a factored form of the equation f(x) = k(x-0)(x-3) provides a straightforward solution. The coefficient k can be any real number, and it scales the parabola without affecting the location of the roots. Students should get accustomed to switching between different forms of quadratic functions and using the best-suited method for solving.