Problem 14 - 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 vertex \((2,3)\)

Step-by-Step Solution

Answer

The set of all parabolae with vertex (2,3) can be represented by the equation \(y = a(x - 2)^2 + 3\), where 'a' is any real number.

Step 1: Vertex Identification

The vertex of the parabola is provided in the problem and is (2,3). This is the point (h, k) in the vertex form of the parabola.

Step 2: Substitute Vertex into Formula

Substitute the values of h = 2 and k = 3 into the standard vertex form equation, \(y = a(x - h)^2 + k\). This gives us the equation \(y = a(x - 2)^2 + 3\).

Step 3: Final Formula Solution

Note that the constant 'a' can be any real number, so the final answer is \(y = a(x - 2)^2 + 3\) where a is a real number. This represents the set of all parabolae with vertex (2,3).

Key Concepts

Equation of a Parabola

When dealing with parabolas, there is a specific way to write their equations, known as the vertex form. The vertex form is extremely useful because it immediately tells us the location of the parabola's vertex. The general structure of the vertex form equation is:\[ y = a(x - h)^2 + k \]Here, \(a\), \(h\), and \(k\) are constants. This equation illustrates how the graph of the parabola is positioned on the coordinate plane.

• \(y\) is the output or the dependent variable (often corresponding to height or distance).
• \(x\) is the independent variable (often representing time or distance).
• \(a\) determines the direction and "width" of the parabola; if \(a\) is positive, the parabola opens upwards and if negative, it opens downwards.
• \((h, k)\) is the vertex of the parabola, providing its precise location on the graph.

By substituting the coordinates of the vertex into this formula, you can derive the specific equation of any parabola with that vertex.

Vertex of a Parabola

The vertex of a parabola plays a critical role in its representation. It is the peak or the lowest point of the graph, depending on whether the parabola opens upwards or downwards.

• For a parabola of the form \(y = a(x - h)^2 + k\), the vertex is directly taken from the coordinates \((h, k)\).
• The vertex provides a central point around which the graph is symmetrical.
• By shifting \(h\) or \(k\), you effectively move the parabola along the x-axis or y-axis without changing its shape or orientation.

In our exercise, the vertex given is (2,3). This means the center of the parabola is at this coordinate, hence moving the entire graph according to this fixed point.

Real Number Coefficients

In mathematics, coefficients are the numbers that multiply the variables in an equation. When we talk about real number coefficients in the context of parabolas in vertex form, we mean that the value of \(a\) can be any real number. Real numbers include all possible values on the number line, except for imaginary numbers.

• The coefficient \(a\) affects the parabola's shape: if \(|a| > 1\), the parabola is narrower; if \(0 < |a| < 1\), it's wider.
• Whether \(a\) is positive or negative determines if the parabola opens upwards or downwards.
• Since \(a\) is a real number, it allows an infinite variety of parabolas, all sharing the same vertex yet differing in shape and orientation based on \(a\).

This flexibility is what makes parabolas with a given vertex versatile when it comes to modeling real-world situations. The parameter \(a\) enables the modeling of phenomena ranging from the trajectory of projectiles to the representation of simple quadratic functions.