Problem 74 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describe a translation of the graph that will result in a function with (a) four distinct real zeros, (b) two real zeros, each of multiplicity \(2,(\mathrm{c})\) two real zeros and two imaginary zeros, and (d) four imaginary zeros.

Step-by-Step Solution

Answer

For (a), move the graph to intersect the x-axis at four different points, for (b) make the graph touch the x-axis at two points but don't let it cross, for (c) shift the graph to cut the x-axis only at two spots and for (d), move the graph entirely above or below the x-axis.

Step 1: Case (a) - Four Distinct Real Zeros

For four distinct real zeros, shift the graph to intersect the x-axis at four different places. Any transformation that achieves this will give four distinct real zeros. It would involve movements both vertically (up or down) and horizontally (left or right).

Step 2: Case (b) - Two Real Zeros, Each of Multiplicity 2

For two real zeros each of multiplicity 2, we need to make the graph touch the x-axis at two points but not cross it. This could be obtained by shifting and stretching the graph. In other words, the graph should be tangent to the x-axis at two points. Any transformation achieving these conditions will give two real zeros of multiplicity 2.

Step 3: Case (c) - Two Real Zeros and Two Imaginary Zeros

This requires the graph to intersect the x-axis at two points (for the two real zeros) and no other x-intercept (for the two imaginary zeros). Shift and/or stretch the graph so it only cuts the x-axis at two points.

Step 4: Case (d) - Four Imaginary Zeros

For the graph to have four imaginary zeros, it should not intersect the x-axis at all. We therefore need to shift the graph either entirely above or entirely below the x-axis. Such a transformation would result in a function whose zeros are all imaginary.

Key Concepts

Real Zeros

Real zeros of a function, also known as real roots or solutions, are the values of the variable that make the function equal to zero. These zeros appear as points where the graph of the function intersects or touches the x-axis.
To find real zeros, you can set the function equal to zero and solve the resulting equation. In simpler terms, if a function is in the form of a polynomial, such as \( f(x) = x^3 - 3x^2 + 3x - 1 \), the real zeros are the x-values where this polynomial equals zero.

• Graphically, real zeros correspond to x-intercepts.
• If a graph makes contact with the x-axis at multiple distinct points, each point corresponds to a distinct real zero.

Understanding where a graph intersects the x-axis helps identify the number and nature of the real zeros, which is crucial for analyzing and predicting the function's behavior.

Imaginary Zeros

While real zeros represent the points where a graph crosses the x-axis, imaginary zeros are the roots that do not result in x-axis intersections. These occur when a function, such as a polynomial, cannot be factored into real numbers. The solutions to these are complex numbers, typically involving the imaginary unit \( i \), which is defined by \( i^2 = -1 \).

A graph with only imaginary zeros does not touch or intersect the x-axis at any point. This feature lets us visually discern when a polynomial has non-real solutions. Imaginary zeros often come in conjugate pairs, such as \( a + bi \) and \( a - bi \), due to the nature of real coefficients in polynomial equations.

• Imaginary zeros result in quadratic factors without real solutions.
• They are evident in polynomial graphs that sit entirely above or below the x-axis.

Employing complex number calculations is essential for solving imaginary zeros, particularly in quadratic equations that don't factor into real solutions.

Multiplicity of Zeros

Multiplicity refers to the number of times a particular zero appears in the context of a polynomial function. When a graph has zeros with higher multiplicities, it tends to "bounce" or "turn" at those x-intercepts instead of crossing the axis.
For example, if a zero of a function is \( x = 2 \) with a multiplicity of 2, the graph will touch the x-axis at this point but not pass through it.

• When multiplicity is odd, the graph crosses the x-axis. When even, it just touches it.
• A zero's multiplicity is indicated by the exponent in the factor form of the polynomial.

This concept helps determine the graph's shape around an intercept and indicates the behavior of the function near particular x-values, making it a valuable tool in sketching graphs and solving polynomial equations.

X-Axis Intersection

X-axis intersections occur where a graph of a function crosses or touches the x-axis. These intersections are the graphical representation of the function's real zeros.
In polynomial functions, every x-axis intersection signifies a real zero. This point corresponds directly to a root of the equation \( f(x) = 0 \)
By examining these intersections, we can gain insight into the function's behavior and characteristics:

• If the graph crosses the x-axis, the zero has an odd multiplicity.
• If the graph merely touches the axis, the zero has an even multiplicity.
• The number of real zeros equals the number of times the graph intersects the x-axis.

Understanding this concept aids in interpreting graphs and predicting the number of roots a polynomial function might have, which is essential for solving equations and analyzing function behavior.