Problem 13 - 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 \(x=-5\) and \(x=1\)

Step-by-Step Solution

Answer

The set of parabolas with x-intercepts at \(x=-5\) and \(x=1\) is represented by the function \(f(x)=a(x+5)(x-1)\), where \(a\) can be any constant number.

Step 1: Identify the x-intercepts

The x-intercepts for this function are given as \(x=-5\) and \(x=1\). These are the points where the parabola crosses the x-axis.

Step 2: Apply the x-intercepts in the formula

Substitute these values into the general form \(f(x)=a(x-h)(x-k)\), we get \(f(x)=a(x+5)(x-1)\).

Step 3: Find the general form of the function

Thus, the general form of the function that represents the set of parabolas with x-intercepts at \(x=-5\) and \(x=1\) is \(f(x)=a(x+5)(x-1)\), where \(a\) can be any constant number.

Key Concepts

Parabolas

Parabolas are important shapes in mathematics and can be observed in graphs of quadratic functions. They have a symmetrical, U-shaped curve that opens either up or down. How a parabola opens depends on the value of the leading coefficient in its equation:

• If the coefficient is positive, the parabola opens upwards.
• If the coefficient is negative, it opens downwards.

Parabolas have critical points called a vertex, which represents the maximum or minimum point of the curve. This point is crucial because it tells us the highest or lowest value of the function. Parabolas also have an axis of symmetry, a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves.
Understanding the features of parabolas helps in graphing and solving quadratic equations. It also aids in identifying the role of each part of the quadratic function in determining the shape and position of a parabola on a graph.

x-intercepts

The x-intercepts of a parabola are the points where the curve crosses the x-axis. These are essential because they reveal the solutions of the quadratic equation when set to zero.
To find x-intercepts:

• Set the quadratic equation equal to zero. For example, \(ax^2 + bx + c = 0\).
• Solve for the values of \(x\) that satisfy this equation.

In the original problem, the x-intercepts are given as \(x = -5\) and \(x = 1\). These intercepts help in constructing the quadratic equation in its factored form: \(f(x) = a(x + 5)(x - 1)\). Knowing the x-intercepts allows us to understand how the parabola will position itself with respect to the x-axis on a graph, giving a clear picture of where the function's values switch from positive to negative or vice versa.

Vertex Form

The vertex form of a quadratic function is a way to express the function that highlights the vertex. It's structured as \(f(x) = a(x - h)^2 + k\), where:

• \(a\) determines the direction and width of the parabola.
• \(h\) and \(k\) represent the vertex's coordinates.

This form is particularly useful when you wish to easily identify the parabola’s vertex without converting from another format. It makes graphing simpler and more intuitive, as you can immediately locate the turning point of the parabola and understand its basic shape. Converting from the general form \(ax^2 + bx + c\) to the vertex form involves completing the square, but having the vertex form directly provides convenience in many contexts.

General Form of a Quadratic Equation

The general form of a quadratic equation is the staple expression of a quadratic function, written as \(ax^2 + bx + c\). Each component of this equation has a specific role in describing the parabola:

• \(a\) determines the parabola's direction (upwards if positive, downwards if negative) and its width.
• \(b\) affects the position of the vertex along the x-axis.
• \(c\) determines the y-intercept, the point where the parabola crosses the y-axis.

From the original exercise, using the x-intercepts \(x = -5\) and \(x = 1\), you derive the factors as \(f(x) = a(x + 5)(x - 1)\). Expanding these factors gives a form that resembles \(ax^2 + bx + c\), showcasing how factors can be used to find intercepts and create a bridge between factored form and general form. Understanding this transition enhances comprehension of how quadratic equations are solved and graphed.