Problem 64 - Algebra 2

Question

a. Compare the graphs of \(y=(x+1)(x+2)(x+3)\) and \(y=(x-1)(x-2)(x-3)\) . What transformation could you use to describe the change from one graph to the other? b. Compare the graphs of \(y=(x+1)(x+3)(x+7)\) and \(y=(x-1)(x-3)(x-7) .\) Does the transformation that you chose in part (a) still hold true? Explain. c. Make a Conjecture What transformation could you use to describe the effect of changing the signs of the zeros of a polynomial function?

Step-by-Step Solution

Answer

The transformation from \(y=(x+c)(x+c_2)(x+c_3)\) to \(y=(x-c)(x-c_2)(x-c_3)\), where \(c, c_2, c_3\) are positive, is a horizontal shift to the right by a distance that is twice the absolute value of the negative zeros.

Step 1: Identify the zeros of the first polynomial

The zeros of the polynomial function given by the equation \(y=(x+1)(x+2)(x+3)\) are obtained by setting each factor equal to zero. This gives the zeros as \(x=-1\), \(x=-2\), and \(x=-3\).

Step 2: Identify the zeros of the second polynomial

Similarly, the zeros of the polynomial function \(y=(x-1)(x-2)(x-3)\) are \(x=1\), \(x=2\), and \(x=3\), found by setting each factor to zero.

Step 3: Describe the transformation for part (a)

The transformation from the graph of \(y=(x+1)(x+2)(x+3)\) to the graph of \(y=(x-1)(x-2)(x-3)\) can be described as a horizontal shift to the right by 4 units. Each zero in the first polynomial is increased by 4 to find the corresponding zero in the second polynomial.

Step 4: Identify the zeros of the third polynomial

For the polynomial \(y=(x+1)(x+3)(x+7)\), the zeros are \(x=-1\), \(x=-3\), and \(x=-7\).

Step 5: Identify the zeros of the fourth polynomial

The zeros of the polynomial \(y=(x-1)(x-3)(x-7)\) are \(x=1\), \(x=3\), and \(x=7\).

Step 6: Check the consistency of the transformation for part (b)

Examining both sets of zeros, we see that each zero from the polynomial in part (b) is shifted to the right by the same amount as before, which is 4 units. Therefore, the transformation described in part (a) does hold true for part (b).

Step 7: Formulate a conjecture for the general transformation

Based on the previous observations, one can conjecture that the transformation for changing the signs of the zeros of a polynomial from positive to negative (or vice versa) is a horizontal shift. If each zero is increased by \(a\), then the shift is \(a\) units to the right. If each zero is decreased by \(a\), then the shift is \(a\) units to the left.

Key Concepts

Zeroes of a Polynomial

The zeroes of a polynomial, also known as its roots, are the values of the variable that make the polynomial equal to zero. For example, to find the zeroes of the polynomial given by the equation y=(x+1)(x+2)(x+3), we need to solve for x when y=0. Each factor set equal to zero gives us a zero of the polynomial—x=-1, x=-2, and x=-3.

Understanding the concept of zeroes is essential for graphing polynomials because these points are where the graph will intersect the x-axis. When a zero is repeated, it indicates a point where the graph bounces off the axis at the intercept or passes through with a flatter curve, depending on the multiplicity of the zero.

Horizontal Shift

A horizontal shift in the context of polynomial transformations is when the entire graph of a polynomial function moves to the left or right along the x-axis. This can be observed by a consistent change in the zeros of the polynomial. For instance, if every zero of a polynomial is increased by a value of a, the graph will shift a units to the right.

In our exercise, the comparison of the polynomials y=(x+1)(x+2)(x+3) and y=(x-1)(x-2)(x-3) reveals a horizontal shift. Each zero of the first polynomial is increased by 4, resulting in the zeros of the second polynomial, hence, the graph shifts 4 units to the right. This fundamental aspect of polynomial graphing is crucial for understanding how changes in the polynomial's equation will affect its graph.

Graphing Polynomials

Graphing polynomials involves plotting the points where the function intersects the x-axis (zeroes of the polynomial) and y-axis, determining the end behavior (how the function behaves as x approaches infinity or negative infinity), and finding the turning points where the graph changes direction.

The polynomial functions from our examples, y=(x+1)(x+2)(x+3) and y=(x-1)(x-2)(x-3), would both yield curves that intersect the x-axis at their respective zeros and will extend infinitely upwards or downwards as x moves away from zero. The horizontal shift between these polynomials allows students to visualize how the graph is transferred along the x-axis without changing its shape.

Conjecturing in Mathematics

Conjecturing is the process of forming a hypothesis or educated guess based on observed patterns or evidence before proving it to be true or false. In mathematics, conjecturing is a starting point for the exploration of patterns and leads to deeper understanding and formal theorems.

In our exercise, after comparing the zeros of different polynomials and observing the horizontal shift, we can make a conjecture. We hypothesize that changing the signs of all zeros of a polynomial produces a consistent horizontal shift across all cases. This conjecture serves as a prediction that can then be tested with different polynomials to determine its validity.