Problem 39 - Precalculus : Building Concepts and Connections

Question

Sketch the polynomial function using transformations. $$h(x)=-2 x^{5}-1$$

Step-by-Step Solution

Answer

The original function \( f(x) = x^{5} \) is vertically stretched by a factor of 2 and then shifted downwards by 1 unit. The resulting graph of \( h(x) = -2x^{5} - 1 \) increases rapidly for positive x and decreases slowly for negative x, with all points centered around \( y = -1 \).

Step 1: Recognize the original function and the transformations

The given polynomial can be written as \( h(x) = -2x^{5} - 1 \). We can think of it as the original function \( f(x) = x^{5} \) which is then affected by two transformations. To break it down, \( -2 \) indicates a vertical stretch by a factor of 2, and \( -1 \) suggests a vertical shift downwards by 1 unit. In other words, each point on the original function is stretched away from the x-axis by a factor of 2, and then moved 1 unit down.

Step 2: Sketch the original function

Start by sketching the original function \( f(x) = x^{5} \). This is a simple polynomial function. For \( x < 0 \), the function is negative, at \( x = 0 \), the function is equal to 0, and for \( x > 0 \), the function is positive. The curve increases slowly for negative x, passes through the origin, and then increases rapidly for positive x.

Step 3: Apply the vertical stretch

Next, apply the vertical stretch. The factor of the stretch is given by the coefficient of \( x^{5} \), which is -2. Multiplying a function by a negative number reflects the graph across the x-axis, and the larger absolute value means that every y-coordinate of the original graph is multiplied by 2. This doesn't change the shape of the curve but makes it steeper.

Step 4: Apply the vertical shift

Finally, apply the vertical shift which is to move the graph downwards by 1 unit. Each point on the stretched graph has its y-coordinate reduced by 1. The effects of this vertical translation are: all points move straight downwards, the curve's general shape does not change, and it is now centered around \( y = -1 \) as opposed to \( y = 0 \).

Step 5: Complete the graph

Complete the sketch of the function by plotting an appropriate scale on both x and y-axis. This results in the final graph of \( h(x) = -2x^{5} - 1 \), which is a vertically stretched and shifted version of \( f(x) = x^{5} \).

Key Concepts

Vertical Stretch

A vertical stretch refers to the transformation of a function where every point on the graph is moved further away from the x-axis. It involves multiplying all original y-values by a constant factor. This transformation either makes the graph steeper or flatter, depending on the value of the factor. In the case of the polynomial function given, \( h(x) = -2x^{5} - 1 \), the factor is 2, which means that for each original point on the graph of \( f(x) = x^{5} \), the y-coordinate is scaled by 2.

It is important to note a few characteristics of vertical stretches:

• A factor greater than 1 "stretches" the graph away from the x-axis, making it steeper.
• If the factor is between 0 and 1, the graph "shrinks", becoming less steep.
• The factor, in this case, also has a negative sign, which additionally causes the reflection of the graph across the x-axis.

Understanding how to interpret such transformations is crucial for correctly sketching or analyzing transformed polynomial functions.

Vertical Shift

A vertical shift is one of the simplest types of transformations applied to a function. Here, you move the entire graph up or down along the y-axis without affecting its shape. For the function \( h(x) = -2x^{5} - 1 \), there is a vertical shift of "-1." This means every point on the graph of the original function \( f(x) = x^{5} \) is moved one unit downward.

Key points to consider about vertical shifts:

• Vertical shifts do not alter the steepness or the orientation of the graph.
• A positive shift raises the graph, while a negative shift lowers it.
• The transformations change the y-intercept of the polynomial by moving it up or down.

This particular transformation helps set the new "center" or baseline where the symmetry or key features of the graph align.

Reflection Across the X-Axis

Reflection across the x-axis is seen as a transformation that flips the entire graph over the horizontal x-axis. This is a result of a sign change in the coefficient of a function. For the polynomial \( h(x) = -2x^{5} - 1 \), reflection is caused by the negative sign in front of the coefficient \( -2 \). This leads to every point on the original function \( f(x) = x^{5} \) being mirrored across the x-axis.

Some detailed aspects of reflections across the x-axis include:

• This transformation keeps the graph's shape intact but alters the direction of the points from above to below the x-axis or vice versa.
• A reflection affects only the y-values, essentially multiplying each by -1.
• Such reflections are critical in functions where direction or orientation has specific significance.

When applied in conjunction with other transformations like stretches or shifts, reflections offer valuable insights into the comprehensive behavior of polynomial graphs.