Problem 24 - Precalculus : Building Concepts and Connections

Question

Identify the underlying basic function, and use transformations of the basic function to sketch the graph of the given function. $$h(x)=-\frac{1}{3}(x-2)^{2}-\frac{3}{2}$$

Step-by-Step Solution

Answer

The function \(h(x) = -\frac{1}{3}(x-2)^{2}-\frac{3}{2}\) is a vertically compressed, reflected quadratic function shifted 2 units to the right and 3/2 units down. The graph will be an open-down parabola.

Step 1: Identify the basic function

The basic function underlying the given function \(h(x)=-\frac{1}{3}(x-2)^{2}-\frac{3}{2}\) is the quadratic function \(f(x) = x^2\). In \(h(x)\), \(x^2\) is being transformed.

Step 2: Determine the transformations

In \(h(x) = -\frac{1}{3}(x-2)^{2}-\frac{3}{2}\), several transformations can be observed: \1. The factor -1/3 reflects the graph about the x-axis and vertically compresses it by a factor of 1/3. \2. The term (x-2) shifts the graph 2 units to the right. \ 3. Finally, subtracting 3/2 from the function moves the graph 3/2 units down.

Step 3: Sketch the graph

Start sketching the graph with the base function \(f(x) = x^2\). \1. Reflect it about x-axis to account for the negative sign, leading to an open-down parabola. \2. The factor 1/3 compresses the graph vertically by that factor. \3. Move 2 units to the right for the (x-2). \ 4. Finally, shift the graph 3/2 units down taking into consideration the -3/2. \The precise details of the graph would require calculating key points, such as the vertex and any x-intercepts, but this gives an overall shape and position for the graph of the function \(h(x)\).

Key Concepts

Quadratic Functions

Quadratic functions form the foundation of many transformations seen in algebra. A basic quadratic function has the form \( f(x) = ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are constants, and the graph of this function is a parabola. Specifically, when you have \( f(x) = x^2 \), it's the simplest form, with a vertex at the origin (0,0) and an axis of symmetry along the y-axis.
Quadratics have certain characteristic features:

• The graph is always a parabola.
• The parabola opens upwards when \( a > 0 \) and downwards when \( a < 0 \).
• The vertex represents the point of minimum or maximum value of the function, depending on whether it opens up or down.

Understanding these basics provides a platform to explore more complex transformations like those seen in the given function \( h(x) = -\frac{1}{3}(x-2)^{2} - \frac{3}{2} \).

Graph Sketching

Graph sketching is about transforming the basic form of functions to visualize them in different ways. With a quadratic function like \( f(x) = x^2 \), starting with the simplest form makes it easier. Let’s break it down step-by-step considering the transformations involved.
First, recognize the base function: in the given example, \( f(x) = x^2 \) is that base function. The job is to apply various transformations to this base function to sketch the graph of \( h(x) \). Here’s how:

• Begin with the basic parabola of \( f(x) = x^2 \).
• Apply any reflections and compressions/stretching along the y-axis.
• Shift the graph horizontally and vertically as required by the transformations.

Accurate graph sketching requires identifying the new vertex, x-intercepts, and y-intercepts, which offer a good picture of the function’s behavior and shape.

Horizontal and Vertical Shifts

Horizontal and vertical shifts change the position of the graph without altering its shape or orientation. In the function \( h(x) = -\frac{1}{3}(x-2)^{2}-\frac{3}{2} \), two shifts are evident.
For horizontal shifts, observe the term inside the parentheses. Here, \( (x-2) \) means a shift of 2 units to the right because \( x \, \rightarrow \, x-c \) shifts the graph "c" units to the right.
Vertical shifts are straightforward. For \( h(x) \), the term \(-\frac{3}{2}\) moves the entire graph 1.5 units downward. Such shifts are direct,

• Horizontal shifts come from changes within the \( x \) variable.
• Vertical shifts result from adding or subtracting a constant outside of \( (x - c)^2 \).

These shifts allow graphs to be positioned in specific parts on the coordinate plane, providing visual insights into function behavior.