Problem 56 - A Graphical Approach to Precalculus with Limits

Question

RELATING CONCEPTS For individual or group investigation (Exercises \(55-58\) ) The concepts of stretching, shrinking, translating, and reflecting graphs presented earlier can be applied to polynomial functions of the form \(P(x)=x^{n}\). For example. the graph of \(y=-2(x+4)^{4}-6\) can be obtained from the graph of \(y=x^{4}\) by shifting 4 units to the left. stretching vertically by applying a factor of \(2,\) reflecting across the \(x\) -axis, and shifting downwand 6 units, so the graph should resemble the graph to the right. If we expand the expression \(-2(x+4)^{4}-6\) algebraically, we obtain $$ -2 x^{4}-32 x^{3}-192 x^{2}-512 x-518 $$ Thus, the graph of \(y=-2(x+4)^{4}-6\) is the same as that of $$ y=-2 x^{4}-32 x^{3}-192 x^{2}-512 x-518 $$ In Exercises \(55-58,\) two forms of the same polynomial finction are given. Sketch by hand the general shape of the graph of the function and describe the transformations. Then support your answer by graphing it on your calculator in a suitable window. $$\begin{array}{l} y=-3(x+1)^{4}+12 \\ y=-3 x^{4}-12 x^{3}-18 x^{2}-12 x+9 \end{array}$$

Step-by-Step Solution

Answer

Shift left 1, reflect over x-axis, stretch by 3, shift up 12.

Step 1: Identify the Base Function

The base function given is \(y = x^4\). This is a basic quartic function (of degree 4) that is symmetric over the y-axis and has a shape similar to a "U".

Step 2: Analyze the Transformed Function

The given function is \(y = -3(x+1)^4 + 12\). Each part of this expression suggests a transformation from the base function \(y = x^4\).

Step 3: Shift the Graph Horizontally

The term \((x+1)\) indicates a horizontal shift of the graph. Specifically, adding 1 inside the function \((x+1)^4\) means the graph is shifted 1 unit to the left.

Step 4: Apply Vertical Stretch and Reflection

The coefficient \(-3\) before \((x+1)^4\) affects the vertical stretch and reflection. Since it is negative, the graph will reflect over the x-axis, and because it is 3, the graph will also be stretched vertically by a factor of 3.

Step 5: Shift the Graph Vertically

The constant term \(+12\) outside of the function \((x+1)^4\) indicates a vertical shift upwards by 12 units.

Step 6: Compare to Expanded Form

The expanded form of the polynomial is \(y = -3x^4 - 12x^3 - 18x^2 - 12x + 9\). Both forms describe the same graph, confirming that the transformations were correctly applied.

Step 7: Sketch the Graph

Based on these transformations, sketch the graph: 1) Start with \(y=x^4\), 2) shift 1 unit left, 3) reflect over the x-axis, 4) stretch vertically by 3, and finally 5) shift vertically upwards by 12 units.

Key Concepts

Graph Transformations

Understanding graph transformations can make analyzing polynomial functions more intuitive. Transformations allow us to take a simple graph and modify it in predictable ways. Here are some key types of graph transformations you might encounter:

• Horizontal Shifts: move the graph side to side.
• Vertical Shifts: move the graph up or down.
• Vertical Stretches/Shrinks: change the graph's height.
• Reflections: flip the graph over an axis.

These transformations are like tools that help you explore what happens to the graph when you change certain parameters in a function. With sufficient practice, identifying these transformations will become easier and help in sketching and understanding graphs efficiently.

Horizontal Shift

A horizontal shift involves moving a graph left or right along the x-axis. The shifting factor is crucial in determining the direction of the shift:
A common form of a polynomial function after a horizontal shift might look like \( f(x) = (x - h)^n\). Here:

• If \(h > 0\), the graph shifts \(h\) units to the right.
• If \(h < 0\), the graph shifts \(|h|\) units to the left.

In our example with \(y = -3(x+1)^4 + 12\), the term \(x + 1\) indicates a shift 1 unit to the left. Though it seems like it should move right, inside the parentheses, the sign is flipped, so the graph moves in the opposite direction you might initially think.

Vertical Stretch

The vertical stretch affects the height of the graph, making it taller or shorter by multiplying the function by a certain factor. The coefficient in front of the function is responsible for this transformation.
In general, if the function looks like \(a \, (x)^n\), where \(a\) is the coefficient:

• If \(|a| > 1\), the graph is stretched vertically by a factor of \(|a|\).
• If \(|a| < 1\), the graph is compressed vertically.

For \(y = -3(x+1)^4 + 12\), the coefficient is -3. We first note that \(|-3| = 3\), indicating a vertical stretch by a factor of 3. This makes the graph's peaks and valleys sharper and more pronounced, effectively magnifying the function's variations.

Reflection

Reflections flip the graph of a function over a particular axis. When a polynomial is multiplied by a negative factor, this reflects the graph over the x-axis.
For any function \(f(x)\), multiplying it by \(-1\) results in a reflection:

• Reflection over the x-axis: The equation becomes \(-f(x)\).
• Reflection over the y-axis: Only occurs when the inside variable changes sign, like \(f(-x)\).

In our case, the \(-3\) factor in \(y = -3(x+1)^4 + 12\) reflects the graph over the x-axis. Horizontal aspects of the graph's shape remain the same, but imagine the graph as a mirror image flipped beneath the x-axis.