Question

\(f(x)=2 x^5-x^3+x^2+4\); reflection in the \(y\)-axis and a vertical stretch by a factor of 3 , followed by a translation 1 unit down

Short answer

The final transformed function is \(g(x)= -6x^5 + 3x^3 + 3x^2 +11\).

Step-by-step solution

Reflection in the y-axis

The reflection in the y-axis transforms the function \(f(x)\) to \(f(-x)\). The new function becomes \(f(-x)=2(-x)^5-(-x)^3+(-x)^2+4 = -2x^5 + x^3 + x^2 + 4.

Vertical Stretch

The vertical stretch by a factor of 3 transforms the function \(f(x)\) to \(3f(x)\). So, the function from Step 1 is then transformed as follows: \[3f(-x)=3(-2x^5 + x^3 + x^2 + 4) = -6x^5 + 3x^3 + 3x^2 +12.\]

Translation 1 unit down

The translation 1 unit down implies subtracting 1 from the function. So the function from Step 2 becomes \[g(x)= -6x^5 + 3x^3 + 3x^2 +12 -1 = -6x^5 + 3x^3 + 3x^2 +11.\] This yields the final transformed function.

Key concepts

Reflection in the y-axis

The concept of reflection in the y-axis is an important transformation within the realm of function manipulation. Imagine looking in a mirror placed along the y-axis of a graph; the reflection of a function, denoted as f(x), across this line results in the new function f(-x). This effectively switches the sign of all the x-terms in the function.

For polynomials, this means that all terms with an odd exponent will change signs, while even exponents will remain the same. For instance, if we have the polynomial function f(x) = 2x^5 - x^3 + x^2 + 4, reflecting it across the y-axis yields f(-x) = -2x^5 + x^3 + x^2 + 4. Through this transformation, the shape of the graph remains consistent, but the sides are effectively swapped, like an image in a mirror.

Vertical Stretch

A vertical stretch alters the steepness or compression of a function's graph without changing its orientation. This transformation multiplies each y-value of the function by a certain factor. Mathematically, this is expressed as cf(x), where c is the stretching factor. A stretch makes the graph taller if c > 1 and squatter if 0 < c < 1.

A vertical stretch does not affect the x-intercepts or the y-intercept, but it does change the distance of the points from the x-axis. For example, applying a vertical stretch by a factor of 3 to our function after the reflection would transform f(-x) into 3f(-x), thereby tripling the 'height' of each point on the graph. This results in the transformed function -6x^5 + 3x^3 + 3x^2 + 12.

Function Translation

Function translation involves sliding a graph vertically or horizontally without distorting its shape. A vertical translation moves the graph up or down, depending on the direction of the shift. This is achieved by simply adding or subtracting a constant to the entire function, denoted by f(x) + k or f(x) - k, where k represents the magnitude and direction of the shift.

For the function we've been transforming, a translation 1 unit down is expressed as f(x) - 1. Applying this last transformation step to our function after the vertical stretch gives us -6x^5 + 3x^3 + 3x^2 + 11, moving every point of the graph down by one unit. Thus, even the horizontal asymptotes and intercepts slide down by this amount, reshaping the function's position on the coordinate plane without affecting the x-intercepts.

Polynomial Transformations

Putting It All Together

When it comes to polynomial transformations, combining reflections, stretches, and translations can drastically change the graph of the original function. Each transformation needs to be applied in the correct order to achieve the desired result. Reflecting polynomials in the y-axis, stretching them vertically, and translating them vertically or horizontally are common operations in algebra and calculus.

A clear understanding of how these transformations impact the graph provides insights into the function's behavior and new ways to analyze real-world situations modeled by polynomials. By mastering these transformations, you can manipulate equations to fit the context of a given problem and gain a deeper understanding of the abstract world of functions.