Question

The graph of the function \(f\) is to be transformed as described. Find the function for the transformed graph. \(f(x)=x^{3}+x-1\); shifted vertically upward by 3 units

Short answer

The transformed function after shifting vertically upwards by 3 units is \(g(x) = x^3 + x + 2\).

Step-by-step solution

Write down the given function

The function is given as \(f(x) = x^3 + x - 1\).

Apply the vertical shift

To shift the function upwards by 3 units, we need to add 3 to the function. So, the transformed function is given by: \[ g(x) = f(x) + 3 = (x^3 + x - 1) + 3 \]

Simplify the transformed function

Combine the constants to simplify the function: \[ g(x) = x^3 + x + 2 \] The transformed function after shifting vertically upwards by 3 units is \(g(x) = x^3 + x + 2\).

Key concepts

Transformation of Functions

Understanding the transformation of functions is critical for students of calculus and those delving into advanced mathematics. Transformations involve shifting, stretching, compressing, or reflecting a 'parent function' to create a new function. The simplest form of transformation is a vertical shift, which moves the graph up or down on the coordinate plane.

When the function is increased by a certain value, as we saw with the polynomial function \( f(x) = x^3 + x - 1 \), we add that value to the whole function. This means every point on the graph of the original function will move up the same amount. To visualize it think of each point of the graph being lifted straight up.

Here's a crucial tip: the transformations do not affect the shape or the orientation of the graph; they simply reposition it. So, if students understand the base graph, they can easily determine how the graph will look after any vertical shift.

Graphing Polynomial Functions

Polynomial functions present a fundamental type of graph that students will encounter repeatedly. Graphs of these functions are smooth and continuous. The polynomial function given in our exercise \( f(x) = x^3 + x - 1 \) is a cubic function and hence its graph will have the shape commonly associated with cubics—a single curve with a region where the function increases indefinitely, and a region where it decreases indefinitely.

When graphing, it's essential to note the coefficients, the degree of the polynomial, and constant terms. For the cubic function in question, its shape is largely determined by the cubic term \( x^3 \)—which ensures one turning point and the end behavior going off to infinity in opposite directions. The linear term \( x \) and the constant term affect the position of the curve on the graph but not its overall shape. Adding a vertical shift will move the entire graph up or down, but leave these fundamental features unchanged.

Vertical Shifts in Calculus

In calculus, vertical shifts are used to adjust the position of graphs without changing their shape. This concept holds true for various function types, including linear, polynomial, exponential, and more. When we apply a vertical shift to the function \( f(x) = x^3 + x - 1 \), adding 3 units moves the entire graph upward while maintaining the general shape of the cubic curve.

It's particularly noteworthy that vertical shifts do not alter the function's derivatives, since the derivative is concerned with the rate of change, not absolute position. Therefore, if students are asked to find a derivative of a vertically shifted function, they can safely ignore the shift when calculating it.

In the context of the given exercise, a vertical shift reveals how we can manipulate functions to fit new criteria while keeping the relationship between variables constant. This is a powerful tool often utilized in calculus to model real-world situations where certain adjustments are needed to the function's graph.