Problem 22 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Sketch the graphs of the following functions. Beneath the sketch of the function, sketch the graph of the derivative. If the graph is the graph of a quadratic, label the coordinates of the vertex of the corresponding parabola; if the graph has corners, label the coordinates of the corners. (a) \(f(x)=-(x+2)^{2}\) (b) \(f(x)=(x-2)^{2}+3\) (c) \(f(x)=(x-2)(x+4)\) (d) \(f(x)=-2(x-2)(x+4)\) (e) \(f(x)=|x+4|+2\) (f) \(f(x)=|2 x|-3\)

Step-by-Step Solution

Answer

The vertex coordinates of the parabolas for \(f(x)=-(x+2)^{2}\), \(f(x)=(x-2)^{2}+3\), \(f(x)=(x-2)(x+4)\), and \(f(x)=-2(x-2)(x+4)\) are (-2,0), (2,3), (1,-2), and (1,-8) respectively. The corners for \(f(x)=|x+4|+2\) and \(f(x)=|2 x|-3\) are at (-4,2) and (0,-3) respectively.

Step 1: Graph the Functions and Identify the Characteristics

We begin with the quadratic functions. Quadratic functions have a general form of \(f(x)=a(x-h)^{2}+k\), where (h,k) is the vertex of the parabola. If the coefficient 'a' is positive, the parabola is U-shaped and if it's negative, the parabola is inverted. For \(f(x)=-(x+2)^{2}\), plot the graph and identify the vertex at (-2,0). For \(f(x)=(x-2)^{2}+3\), do the same and identify the vertex at (2,3). For \(f(x)=(x-2)(x+4)\) and \(f(x)=-2(x-2)(x+4)\), expand the polynomial, set the derivative equal to zero, solve for x, and find the corresponding y values to identify the vertex and sketch the graph. As for the absolute functions, \(f(x)=|x+4|+2\) and \(f(x)=|2 x|-3\), draw a V-shaped graph. For \(f(x)=|x+4|+2\), shift 4 units to the left and 2 units up, and for \(f(x)=|2 x|-3\), stretch the graph by a factor of 2 and shift down by 3 units.

Step 2: Compute the Derivatives and Graph their Functions

The derivative of a function gives the rate of change of the function at a specific point. For the quadratic functions, compute the derivative using the power rule, \(d/dx[x^n] = nx^{n-1}\), and the chain rule, \(d/dx[f(g(x))] = f'(g(x))g'(x)\). For the absolute value functions, we have to split them at the corner and compute the derivatives of the respective linear functions separately. Once the derivative functions are computed, sketch their graphs.

Step 3: Label the Coordinates of the Vertex and Corners

For the quadratic functions, their vertex is labelled. For the absolute value functions, their corners are the points where the function changes direction. For \(f(x)=|x+4|+2\), we see a corner at (-4,2), and for \(f(x)=|2 x|-3\), the corner is at (0,-3). Label these points on the functions' graphs, and for the derivative functions, demonstrate a break at these points by indicating an open circle.

Key Concepts

Quadratic Functions

Understanding quadratic functions is essential in calculus graphing because they form the simplest polynomial function with interesting properties like vertex and symmetry. These functions take the form \( f(x) = a(x-h)^2 + k \). The constants \( h \) and \( k \) determine the vertex's position, a crucial point in parabolas.

• When \( a > 0 \), the parabola opens upwards, resembling a U shape.
• When \( a < 0 \), it opens downwards, forming an inverted U shape.

By changing the values of \( a \), \( h \), and \( k \), the parabola's width, direction, and specific position vary. For example, the function \( f(x) = -(x+2)^2 \) has its vertex at \(-2,0\), indicating a parabola opening downward.

Derivative Calculations

Calculating the derivative of a function is a way of finding its rate of change or the slope of the tangent line at any given point. For quadratic functions, the derivative is determined using basic calculus rules:

• Power Rule: \( \frac{d}{dx}[x^n] = nx^{n-1} \)
• Chain Rule: \( \frac{d}{dx}[f(g(x))] = f'(g(x))g'(x) \)

For instance, given \( f(x) = -(x+2)^2 \), applying these rules yields a derivative of \( f'(x) = -2(x+2) \). This derivative function shows how quickly the parabola is "falling" at each point, which is vital for graphing the slope of the curve.

Absolute Value Functions

Absolute value functions involve expressions where the absolute value operator \(| \,|\) affects how they behave on a graph, resulting in a V-shaped graph. For example, the function \( f(x)=|x+4|+2 \) produces a sharp corner where the direction of the line changes.

• The basic transformation includes translating and scaling the graph.
• For \( f(x)=|x+4|+2 \), the graph shifts left by 4 units and up by 2 units.
• Similarly, \( f(x)=|2x|-3 \) means a vertical stretch by a factor of 2 and a downward shift by 3 units.

The challenge with these functions lies in splitting them around their corner or point of sharp change, which influences their derivative calculation.

Vertex Identification

Identifying the vertex is vital in understanding quadratic functions because it represents the highest or lowest point of a parabola. It acts as a guide for graphing and finding function properties such as the axis of symmetry.

• In \( f(x)=a(x-h)^2+k \), the vertex is always \((h,k)\).
• It determines the shape and symmetry of a parabola.
• In graphs, it's the "peak" or "valley." For instance, the vertex of \( f(x)=(x-2)^2+3 \) is \((2,3)\).

Identifying and labeling the vertex on a graph helps in drawing a more accurate parabola, allowing for correct interpretation of the graph's behavior.

Graph Sketching

Sketching graphs is an essential skill in understanding the behavior of functions visually. It lets us see the relationships between different aspects like the rate of change or position of vertices.

• Always start by identifying key points such as vertices for parabolas or corners for absolute value functions.
• For quadratic functions, knowing the vertex helps in determining the symmetry and direction of the parabola.
• The derivative's graph shows the slope at different points, important for understanding the change of rate in the function.

For example, sketching the graph of \( f(x)=-(x+2)^2 \) involves plotting its vertex at \(-2,0\) and drawing the parabola downward. These visual tools clarify abstract concepts by providing a tangible representation.