Question

Let \(f(x)=x^3-4 x^2+6\). Write a rule for \(g\). Describe the graph of \(g\) as a transformation of the graph of \(\boldsymbol{f}\). $$ g(x)=-f(x-1)+6 $$

Short answer

The rule for \(g(x)\) is \(g(x) = -x^3+ 7x^2 -11x +5\). The graph of \(g(x)\) can be described as a transformation of the graph of \(f(x)\) in that it is a reflection of \(f(x)\) in the x-axis, shifted 1 unit to the right.

Step-by-step solution

Express g(x) in terms of f(x)

Replace \(f(x-1)\) in \(g(x)= -f(x-1)+6\) by \(f(x) = x^3-4x^2+6\). Here, replace \(x\) in \(f(x) = x^3-4x^2+6\) by \(x-1\) which is the input of f in the function g. We compute \(f(x-1) = (x-1)^3-4(x-1)^2 + 6\).

Simplify the resulting expression from Step 1

Now distribute the minus sign from the function g to our computed value of \(f(x-1)\) from step 1 to obtain:\n sine \(g(x) = -(x-1)^3 + 4(x-1)^2 -6 +6\) which simplifies to \(g(x) = -(x^3-3x^2+3x-1) + 4(x^2-2x+1) -6 +6\) and further simplify this to obtain \(g(x) = -x^3 + 3x^2 -3x +1 +4x^2 -8x +4\). Collecting like terms, we obtain: \(g(x) = -x^3+ 7x^2 -11x +5\).

Identify the transformation

Now we identify the transformations that the function f has undergone to obtain the function g. We can observe that the function g has been reflected about the x-axis (due to the negative sign before the x^3 term in g), and shifted 1 unit to the right (due to the subtraction of 1 from x in the original function f in g). The function g can be described as a vertical reflection and horizontal shift of f.

Key concepts

Algebra 2

In Algebra 2, one of the key skills students must master is the ability to work with and understand various types of functions, including polynomial functions. Algebra 2 extends the algebraic concepts learned in Algebra 1 by including more complex functions and transformations. Understanding these concepts is crucial for progressing in mathematics, including pre-calculus and calculus, as well as applications in science and engineering.

Polynomials, which are expressions consisting of variables and coefficients, are manipulated using various algebraic operations such as addition, subtraction, multiplication, and factoring. In Algebra 2, students learn how to analyze the structure of polynomials, graphical behavior, and how to apply transformations to functions to produce shifts, stretches, and reflections on a graph.

Function Transformation

Function transformation is a core concept in Algebra that involves changing the position, shape, or size of a graph of a function. Transformations can be applied to any function, including linear, quadratic, polynomial, and trigonometric functions, to name a few. The basic types of transformations are:

• Translations: Shifting a graph horizontally or vertically.
• Reflections: Flipping a graph across an axis.
• Stretches and Shrinks: Changing the graph’s size vertically or horizontally.
• Rotations: (less common in basic Algebra) Rotating the graph around a point.

For instance, when we take a polynomial function and add a constant to it, the graph will shift vertically. Similarly, if we multiply the function by -1, it will reflect across the x-axis.

Polynomial Functions

Polynomial functions are algebraic expressions that involve sums of powers of a variable. They are an important class of functions with diverse applications in fields such as physics, economics, and engineering. A polynomial function has the general form:\[ P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0 \]where \( n \) is a non-negative integer and the coefficients \( a_0, a_1, ..., a_n \) are real numbers with \( a_n eq 0 \). The degree of the polynomial is the highest power of the variable, which dictates the behavior of the graph, particularly the number of turns and the end behavior as \( x \) approaches infinity.

In the context of transformations, recognizing how the coefficients and terms affect the graph's shape makes it easier for students to predict how a transformation like translation or reflection will affect the function's graph.