Problem 4 - Precalculus Mathematics for Calculus

Question

The graph of \(f(x)=-2(x-3)^{2}+5\) is a parabola that opens _________ , with its vertex at (_______, ______) and \(f(3)=\) ________ is the (minimum/maximum)_________ value of \(f\)

Step-by-Step Solution

Answer

The parabola opens downwards, with vertex at (3, 5) and maximum value 5.

Step 1: Identify Parabola Orientation

Examine the coefficient of the squared term. Since the equation is given by \(f(x) = -2(x-3)^2 + 5\), notice that the coefficient of \((x-3)^2\) is \(-2\). This negative coefficient implies the parabola opens downwards.

Step 2: Find the Vertex

The vertex form of a parabola is given by \(f(x) = a(x-h)^2 + k\). In this equation, \(h=3\) and \(k=5\), which makes the vertex (h, k) or (3, 5).

Step 3: Evaluate f(3)

Substitute \(x = 3\) into the function \(f(x) = -2(x-3)^2 + 5\). Calculate: \(f(3) = -2(3-3)^2 + 5 = -2(0)^2 + 5 = 5\).

Step 4: Determine Maximum/Minimum Value

Since the parabola opens downwards, it has a maximum value at the vertex. The value of \(f(3)\), which is 5, is the maximum value of the function \(f\).

Key Concepts

Vertex Form

The vertex form of a quadratic function is a powerful way to express equations of parabolas. It is particularly useful in identifying key properties of the parabola at a glance. The general equation for this form is given by \(f(x) = a(x-h)^2 + k\). In this form:

• \(a\) is the coefficient that indicates the direction of the parabola. If \(a\) is negative, the parabola opens downward; if positive, it opens upward.
• \(h\) and \(k\) represent the coordinates of the vertex, \((h, k)\). These values tell us the highest or lowest point of the parabola.

This form is particularly user-friendly for functions like \(f(x) = -2(x-3)^2 + 5\), where it's immediately clear that the vertex of the parabola is at the point \((3, 5)\). By simply examining the numbers, you can determine both the direction in which the parabola opens and its vertex position, making it easy to sketch or analyze the graph.

Quadratic Function

A quadratic function is a type of polynomial that can be represented by the equation \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The parabola is the graph of a quadratic function, and it has a characteristic U-shape.

• The sign of the leading coefficient \(a\) determines whether the parabola opens upward or downward. Positive \(a\) opens upward, while negative \(a\) opens downward.
• The parabola is symmetrical around its vertical line of symmetry that passes through the vertex.

Quadratic functions are among the simplest forms of polynomial functions, and they appear frequently in various contexts such as physics for modelling trajectories, finance for optimizing revenue or costs, and in natural phenomena modeling.

Maximum Value

In the context of a quadratic function represented in vertex form, determining the maximum or minimum value is straightforward. For a downward-opening parabola, this maximum occurs at its vertex. The vertex \((h, k)\) indicates this maximum value. For instance, with the quadratic function \(f(x) = -2(x-3)^2 + 5\), the vertex is \((3, 5)\). This means the function has a maximum value of 5 at \(x = 3\). Due to the downward opening indicated by the negative \(a\), the entire graph curves opening downwards, peaking exactly at the vertex. Such maximum values are particularly important in real-world situations where optimal solutions and maximum outputs are needed. They can help determine crucial turning points or peak values which might denote maximum profit, highest point of flight, or other maxima in various applications.