Problem 31 - Algebra 2 and Trigonometry

Question

Let \(\mathrm{f}(x)=x^{3}+3 x^{2}-2 x-6\) and \(\mathrm{g}(x)=2 \mathrm{f}(x)=2 x^{3}+6 x-4 x-12\) be two cubic polynomial functions. a. How does the graph of \(\mathrm{f}(x)\) compare with the graph of \(\mathrm{g}(x) ?\) b. How do the roots of \(\mathrm{f}(x)\) compare with the roots of \(\mathrm{g}(x) ?\) c. In general, if \(\mathrm{p}(x)=a \mathrm{q}(x)\) and \(a>0,\) how does the graph of \(\mathrm{p}(x)\) compare with the graph of \(\mathrm{q}(x) ?\) d. How do the roots of \(\mathrm{p}(x)\) compare with the roots of \(\mathrm{q}(x) ?\)

Step-by-Step Solution

Answer

The graph of \( g(x) \) is a vertical stretch of \( f(x) \) by a factor of 2; both have the same roots.

Step 1: Understanding the Functions

The functions given are \( f(x) = x^3 + 3x^2 - 2x - 6 \) and \( g(x) = 2f(x) \). Therefore, \( g(x) = 2(x^3 + 3x^2 - 2x - 6) = 2x^3 + 6x^2 - 4x - 12 \).

Step 2: Finding the Graph Relationship

To compare the graphs of \( f(x) \) and \( g(x) \), consider that \( g(x) = 2f(x) \) represents a vertical stretch of the graph of \( f(x) \) by a factor of 2. Thus, the shape of \( f(x) \) and \( g(x) \) remain the same but \( g(x) \) will have its y-values doubled.

Step 3: Comparing the Roots of f(x) and g(x)

The roots of a function occur where the function equals zero. For both \( f(x) \) and \( g(x) \), the roots are in the same place since both reduce to the same equation when solved for zero, \( 2f(x) = 0 \) is the same as \( f(x) = 0 \). Thus, the roots of \( f(x) \) and \( g(x) \) are identical.

Step 4: General Graph Relationship for p(x) = aq(x)

In general, if \( p(x) = aq(x) \) where \( a > 0 \), the graph of \( p(x) \) is a vertical stretch or compression of the graph of \( q(x) \) by a factor of \( a \). If \( a > 1 \), it's a stretch; if \( 0 < a < 1 \), it's a compression.

Step 5: General Root Comparison for p(x) and q(x)

The roots of \( p(x) = aq(x) \) remain the same as those of \( q(x) \) because the constant \( a \) only affects the vertical scaling, and does not shift or alter the x-intercepts of \( q(x) \).

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. They are among the most important and well-studied functions in mathematics. A cubic function, like the ones given in our exercise, is a polynomial of degree three. The general form is \( ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants, and \( a eq 0 \).

• Quadratic functions are polynomials of degree two.
• Cubic functions have an "S" shaped graph, typically featuring one point of inflection.

The given polynomial functions in this exercise are \( f(x) = x^3 + 3x^2 - 2x - 6 \) and \( g(x) = 2f(x) = 2x^3 + 6x^2 - 4x - 12 \). Both \( f(x) \) and \( g(x) \) being cubic functions share similar characteristics in terms of their overall shape. However, because \( g(x) \) is a vertically stretched version of \( f(x) \), it expands differently along the y-axis.

Graph Transformations

Graph transformations involve modifying a graph in several ways such as translating, reflecting, stretching, or compressing. Understanding transformations is crucial for graphically interpreting polynomial functions. Specifically, for the transformation observed in this exercise:

• Vertical Stretch: When \( g(x) = 2f(x) \), the graph stretches vertically. Each y-value on the graph of \( f(x) \) is doubled, forming \( g(x) \). This maintains the symmetry and inflection of the original shape but increases the amplitude of the graph along the y-axis.

Another important transformation that can occur is a horizontal shift or stretch, but in our context, only vertical aspects are affected due to the constant multiplier being applied outside the function.

Roots of a Function

In mathematics, the roots or zeroes of a function are the values of \( x \) which make the function equal to zero. For a polynomial function like \( f(x) \), the roots are the x-values where the graph intersects the x-axis.

• To find the roots of \( f(x) \), solve for \( f(x) = 0 \).
• The roots of \( g(x) = 2f(x) \) are the same, because scaling by a factor does not shift the x-intercepts, merely stretches the graph vertically.

Both \( f(x) \) and \( g(x) \) will have roots at the same x-values because solving \( 2f(x) = 0 \) simplifies to \( f(x) = 0 \), making the roots identical.

Vertical Stretch

A vertical stretch is a transformation that scales the graph of a function away from the x-axis by a factor greater than one. In our exercise, \( g(x) = 2f(x) \) demonstrates a vertical stretch of \( f(x) \) by a factor of 2.

• This transformation ensures that every y-value of \( f(x) \) is multiplied by 2 to form \( g(x) \).
• The x-values or roots remain unchanged, which means the locations of where the graph crosses the x-axis stay the same.
• If the factor were less than 1, it would be considered a vertical compression.

Understanding vertical stretches helps in predicting how modifications in function expressions affect their graphical representations, enabling more intuitive analysis of function behavior.