Problem 67 - Algebra and Trigonometry

Question

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. $$ P(x)=x^{4}+c ; \quad c=-1,0,1,2 $$

Step-by-Step Solution

Answer

Changing \(c\) shifts the graph vertically.

Step 1: Identify the Polynomial Function

We are given a family of polynomial functions with the form \(P(x) = x^4 + c\). We need to graph this polynomial for different values of \(c\): \(-1, 0, 1, \) and \(2\).

Step 2: Choose an Appropriate Viewing Rectangle

To display these graphs clearly, select a viewing rectangle that appropriately captures the behavior of the fourth-degree polynomial. A good choice might be \([-3, 3]\) for the x-axis and \([-5, 10]\) for the y-axis to ensure you can see the effect of different \(c\) values.

Step 3: Graph for \(c = -1\)

Plot the graph of \(P(x) = x^4 - 1\). This graph will be similar to the graph of \(x^4\) but shifted downward by 1 unit. The vertex of the graph will be at \((0, -1)\).

Step 4: Graph for \(c = 0\)

Plot the graph of \(P(x) = x^4\). This is the original fourth-degree polynomial centered at the origin with a vertex at \((0, 0)\). It is symmetric about the y-axis and opens upwards.

Step 5: Graph for \(c = 1\)

Plot the graph of \(P(x) = x^4 + 1\). This is the graph of \(x^4\) shifted upward by 1 unit, with the vertex at \((0, 1)\).

Step 6: Graph for \(c = 2\)

Plot the graph of \(P(x) = x^4 + 2\). This graph is like \(x^4\), but shifted upward by 2 units, with a vertex at \((0, 2)\).

Step 7: Analyze the Effect of Changing \(c\)

As \(c\) changes from \(-1\) to \(2\), the graph of \(P(x) = x^4 + c\) shifts vertically. Decreasing \(c\) moves the graph downward, and increasing \(c\) moves the graph upward. The basic shape of the graph remains the same, as \(x^4\) is symmetric about the y-axis and opens upwards.

Key Concepts

Graphing Polynomials

Polynomial functions like \[P(x)=x^4+c\]often have distinctive graphs, characterized by smooth curves and symmetrical shapes. For this exercise, you're looking at a fourth-degree polynomial, which can be sketched by considering its general structure. The graph of a basic function like \(x^4\)is shaped like a U and is symmetrical about the y-axis. It opens upwards because it is a function with an even highest degree.When graphing polynomials like this for various constants \(c\), it’s crucial to select a proper viewing rectangle. This ensures you can clearly observe and compare the transformations these constants cause. A good choice would be

• x-axis: [-3, 3]
• y-axis: [-5, 10]

This range allows a clear representation of how these graphs move up and down with different values of \(c\). It emphasizes both symmetry and the upward open direction, allowing for a comprehensive comparison.

Effects of Constants on Graphs

The constant \(c\) in a polynomial function, specifically in our case \[P(x) = x^4 + c\], plays a critical role in influencing the function's graph. The basic shape is set but observe what happens with each variation of \(c\):

• When \(c = -1\), the polynomial graph shifts downward by 1 unit. This is because you subtract one from every y-value of the graph \(x^4\).
• For \(c = 0\), this represents the standard graph of the function \(x^4\), centered directly on the origin (vertex at \((0, 0)\)).
• As \(c\) increases to \(1\) and \(2\), each graph is shifted upward by the respective units, representing \(x^4 + 1\) and \(x^4 + 2\). The vertex moves to \((0, 1)\) and \((0, 2)\).

These constants do not change the symmetry or the opening direction of the graph, but only affect the vertical position. The graph simply moves along the y-axis, staying systematically intact.

Shifts in Graphs

In mathematics, understanding how graphs shift can greatly enhance your comprehension of function transformations. For polynomial graphs defined as \[P(x) = x^4 + c\],the value of \(c\) causes vertical shifts. Vertical shifts occur when the entire graph moves up or down, directly altering the vertex's y-coordinate. In our polynomial:

• Negative \(c\) values cause downward shifts. For \(c = -1\), the graph descends one unit vertically.
• Positive \(c\) values cause upward shifts. For instance, \(c = 1\) and \(c = 2\) elevate the graph by 1 and 2 units, respectively.

It's beneficial to use these principles to predict the graph of similar polynomials. Such vertical movements are direct results of adding or subtracting a constant term. Importantly, these shifts do not affect the width, shape, or symmetry of the graph, which maintains its original structure.