Problem 76 - College Algebra

Question

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. $$ q(x)=\left(\frac{1}{4} x\right)^{3}+1 $$

Step-by-Step Solution

Answer

The function is a horizontally stretched and vertically shifted cubic function.

Step 1: Identify the Toolkit Function

The given function is a transformation of a basic toolkit function. The core toolkit function here is the cubic function, described by \( f(x) = x^3 \). This will be our starting point to analyze any transformations.

Step 2: Recognize Horizontal Stretch/Compression

The function \( q(x) = \left( \frac{1}{4} x \right)^3 \) indicates a horizontal stretch. The expression \( \frac{1}{4} x \) means we are stretching the graph horizontally by a factor of 4, since \( \frac{1}{4} \) is less than 1.

Step 3: Identify Vertical Shift

The addition of 1 to the function, \( q(x) = \left( \frac{1}{4} x \right)^3 + 1 \), represents a vertical shift. This means the entire graph of the function will shift up by 1 unit.

Step 4: Sketch the Graph

Begin by sketching the basic cubic function \( f(x) = x^3 \). Then, apply the horizontal stretch by extending the graph sideways by a factor of 4. Finally, shift the entire graph up by 1 unit. This will result in the transformed graph for \( q(x) \).

Key Concepts

Horizontal Stretch

A horizontal stretch transforms the graph of a function along the x-axis. It happens when a function in the form of \( q(x) = f(kx) \) has a factor \( k \) that is a fraction (or less than 1). In this context, it essentially means that each point on the graph moves further from the y-axis. For our specific function, \( q(x) = \left( \frac{1}{4} x \right)^3 + 1 \), the presence of \( \frac{1}{4} x \) indicates a horizontal stretch.

• This is because \( \frac{1}{4} \) is less than 1.
• The graph of the basic function \( f(x) = x^3 \) is stretched sideways by a factor of 4.

When dealing with horizontal stretches, remember to think of it as expanding the graph along the x-axis.The unit distances become spread out, effectively slowing down the slope to seem less steep off the origin.

Vertical Shift

Vertical shifts in functions refer to moving the entire graph of a function up or down on the y-axis. This occurs when a constant is added or subtracted from the function. In our function \( q(x) = \left( \frac{1}{4} x \right)^3 + 1 \),

• The '+1' at the end of the function indicates a vertical shift.
• Every point on the graph is moved up by 1 unit.

Imagine taking the entire graph of \( q(x) = \left( \frac{1}{4} x \right)^3 \) and lifting it straight upwards along the y-axis.This shift happens equally for all x-values without changing the shape of the function.

Graph Sketching

Graph sketching is a useful skill for visualizing how transformations affect basic functions. It combines identifying transformations and physically plotting them on a coordinate plane. For a cubic function like \( f(x) = x^3 \), you begin with the basic shape:

• The curve starts at the origin (0,0), extends upwards in the right direction, and downwards to the left.

Now, apply the transformations:

• First, incorporate the horizontal stretch by spreading the points wider apart along the x-axis.
• Then, perform the vertical shift. Shift each point of this spread-out curve up by 1 unit.

Sketching these steps accurately helps in understanding how multiple transformations together reshape the graph.It's important to adjust the grid scales on your sketch to accommodate stretched and shifted axes properly.