Problem 70 - Algebra and Trigonometry

Question

\(69-74\) . Graph the family of polynomials in the same viewing rectangle, using the given values of \(c\) . Explain how changing the value of \(c\) affects the graph. $$ P(x)=(x-c)^{4} ; \quad c=-1,0,1,2 $$

Step-by-Step Solution

Answer

Changing \( c \) translates the graph horizontally along the x-axis.

Step 1: Understanding the Function

We are given the polynomial function \( P(x) = (x-c)^4 \) for different values of \( c \). This function represents a family of polynomials, in which the value of \( c \) shifts the graph left or right.

Step 2: Identifying the Graphs

Determine the specific polynomial functions for each given value of \( c \):- For \( c = -1 \), \( P(x) = (x+1)^4 \)- For \( c = 0 \), \( P(x) = x^4 \)- For \( c = 1 \), \( P(x) = (x-1)^4 \)- For \( c = 2 \), \( P(x) = (x-2)^4 \)

Step 3: Plotting the Graphs

Graph each of the identified polynomials in the same viewing rectangle. Notice that these graphs are vertically symmetric and each has its vertex at \( x = c \).

Step 4: Analyzing the Effects of \( c \)

Changing the value of \( c \) translates the graph horizontally along the x-axis. Specifically, increasing \( c \) moves the graph to the right, while decreasing \( c \) moves it to the left.

Key Concepts

Shifting Graphs

Shifting graphs is a fundamental concept in understanding how changes in function equations affect the location of their graphs on a coordinate plane. With polynomial functions like the one given, shifting occurs when a constant value is added or subtracted from either the input or the output of the function. In our exercise, we explored the family of polynomials represented by the function \( P(x) = (x-c)^4 \).
This specific type of shift is horizontal, affecting the position of the graph along the x-axis. When you change the value of \( c \), you essentially translate the entire graph either to the left or the right. The cool part is that while the graph moves side to side, its shape remains unchanged. So if you imagine the graph as a rigid structure, altering \( c \) is like dragging this structure horizontally across the grid.
The power of understanding shifts lies in predicting how different equations will look without having to plot points manually. If you grasp the basics of shifting, you can quickly sketch or recognize complex graph behaviors just by looking at the equation.

Graphing Polynomials

Graphing polynomials is an exciting way to visualize mathematical functions and understand their behavior. Polynomials, like \( P(x) = (x-c)^4 \), have various degrees which dictate their general shape and curvature. A polynomial's degree is the highest power of \( x \) in its expression. In our case, the degree is 4, which means the graph will show up as a symmetric curve generally resembling a 'W' shape.
When graphing polynomial functions, certain key features are crucial:

• Identify the degree to understand the basic shape.
• Plot the vertex, which in our function is at \( x = c \).
• Recognize that higher-degree polynomials tend to have more complex turning points and intercepts.

For the given polynomial, plotting different values of \( c \) shows us a family of graphs that look similar but are just positioned differently along the x-axis. As each polynomial's expression derives from a common form, their symmetric nature makes them relatively simple to graph once initial patterns are understood.

Horizontal Translation

Horizontal translation is an elegant way of describing how graphs move left or right on the Cartesian plane. Specifically, it involves altering the input values of a function, which controls where the function's graph appears horizontally. In polynomial functions like \( P(x) = (x-c)^4 \), the horizontal translation is dictated by the constant \( c \).
Here’s how it works:

• If \( c \) is increased, the graph shifts to the right. This is because the vertex of the graph moves to \( x = c \).
• If \( c \) is decreased, the graph shifts to the left. The content of each graph remains identical, just repositioned horizontally.

Understanding horizontal translations empowers you to manipulate equations easily to fit specific graphing needs. By adjusting \( c \), you can place the graph just where you need it, illustrating powerful applications in modeling real-world scenarios or solving practical problems. This simple manipulation of terms offers a great deal of flexibility in function analysis and graphical representation.