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 \(\boldsymbol{f}\).$$ g(x)=f(-x)+4 $$

Short answer

The function \(g(x) = -x^3 - 4x^2 + 10\) is a reflection of the given function \(f(x) = x^3 - 4x^2 + 6\) over the y-axis, followed by a vertical shift upward by 4 units.

Step-by-step solution

Define the Function \(g(x)\)

The given function \(g(x)\) is defined in terms of the function \(f(x)\), so first we need to figure out what \(f(-x)\) is and then add 4 to find \(g(x)\). Substitute \(x\) with \(-x\) into the equation for \(f(x)\) to get \(f(-x) = (-x)^3 - 4(-x)^2 + 6\). Simplify this to \(f(-x) = -x^3 - 4x^2 + 6\). Then add 4 to \(f(-x)\) to find \(g(x) = -x^3 - 4x^2 + 6 + 4 = -x^3 - 4x^2 + 10\).

Analyze the Effect of the Transformation on the Graph

The substitution of \(x\) with \(-x\) reverses the direction of \(x\), resulting in a reflection of the graph over the y-axis. This changes the behavior of the function but retains its shape. The addition of 4 to the function, on the other hand, provides a straight upward shift of the graph by 4 units.

Describe \(g\) as a Transformation of \(f\)

The function \(g(x) = -x^3 - 4x^2 + 10\) can be obtained from the function \(f(x)\) by two transformations: a reflection in the y-axis, accomplished by replacing \(x\) with \(-x\) in \(f(x)\), and a vertical shift upward by 4 units, accomplished by adding 4 to \(f(-x)\). So, the graph of \(g\) is the graph of \(f\), reflected over the y-axis and then shifted up by 4 units.

Key concepts

Reflection over the Y-Axis

Understanding the reflection over the y-axis requires us to visualize how a function behaves when its input variable, often denoted as 'x', is negated. When we apply this transformation to a function, we essentially 'flip' its graph across the y-axis. It's like looking at the function in a mirror placed along the y-axis.

For our exercise, the function notation changes from focusing on the original function, which is a cubic function described by the relation of its input 'x' to its output 'f(x)', to looking at \(f(-x)\). This key step reverses every horizontal movement on the graph, thereby swapping the left and right sides. So if we originally had a curve that for a positive 'x' value curved upwards to the right, after the reflection, it would curve upwards to the left for the same 'x' magnitude but in negative form.

Vertical Shift

A vertical shift is one of the simpler transformations that can be applied to a function's graph. It moves every point on the graph straight up or down along the y-axis. Add a positive number to the original function, and the graph goes up; subtract a number, and it goes down.

This operation is visually straightforward and doesn't change the shape or orientation of the graph; it just repositions it vertically. In the given exercise, we're adding \(4\) to the entire function after the reflection over the y-axis, which means every point on the reflected graph gets lifted up 4 units higher than before.

Function Notation

Function notation is a concise way to define the output of a function based on its input. The most common notation is \(f(x)\), which says 'the function \(f\) evaluated at \(x\)'. This means we're looking at what \(f\) does specifically when we put \(x\) into it.

When dealing with transformations, we often see modifications to this input, as in our exercise with \(g(x) = f(-x) + 4\). Here, we are not just using the function \(f\) but also transforming it in specific ways to obtain the function \(g\). The notation guides us through the steps needed to evaluate the new function at any point along its domain.

Graphing Transformations

Graphing transformations involves modifying a function's graph through various operations such as translations, reflections, stretching, and shrinking. By knowing the nature of the transformation, we can predict and illustrate how the original graph will be altered.

A useful strategy is to observe the effects on the main features of a graph, like its intercepts, maxima, minima, and asymptotes. In our given example, by first reflecting over the y-axis and then shifting the graph vertically upward, we are performing two separate transformations that change the position and orientation of the original cubic graph of \(f(x)\) to obtain the new graph of \(g(x)\). Each transformation follows specific rules, and when combined, they provide a step-by-step map to the final graph from the original.