Problem 28 - Algebra and Trigonometry

Question

\(27-32\) : A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=x^{3} ;\) shift downward 1 unit and shift 4 units to the left

Step-by-Step Solution

Answer

The final transformed equation is \(g(x) = (x + 4)^3 - 1\).

Step 1: Understand Function Transformations

We start with the function \(f(x) = x^3\). When you shift this function downward by 1 unit, it means you subtract 1 from the function: \(f(x) = x^3 - 1\). Next, to shift the function 4 units to the left, you replace every \(x\) in the function with \(x + 4\).

Step 2: Apply Downward Shift Transformation

For the first transformation, subtract 1 from the entire function \(f(x)\):\[ g(x) = x^3 - 1 \]This represents the graph of \(f(x)\) shifted down by 1 unit.

Step 3: Apply Horizontal Shift Transformation

To shift the function 4 units to the left, substitute \(x\) with \(x+4\) in the function \(g(x) = x^3 - 1\):\[ g(x) = (x+4)^3 - 1 \]This completes the transformation: first down by 1 unit and then 4 units to the left.

Key Concepts

Vertical Shift

In transformations, a vertical shift moves each point of a graph up or down on the y-axis by a certain number of units. It affects the output value of a function directly.

When you see '+1' or '-1' added or subtracted from the function, it signals a vertical shift.

• A positive number indicates a shift upwards.
• A negative number indicates a shift downwards.

For instance, if you have a function such as \(f(x) = x^3\) and you apply a vertical shift downward by 1 unit, the new function becomes \(f(x) - 1 = x^3 - 1\).

This transformation lowers the graph along the y-axis, effectively changing every y-coordinate of the original graph by reducing it by 1. So, if the original point was \((x, y)\), it would now be \((x, y-1)\). Understanding these simple shifts is key to visualizing how graphs change with different transformations.

Horizontal Shift

In the realm of transformations, a horizontal shift adjusts the graph along the x-axis. This affects the argument (input) of the function.

When you replace \(x\) in the function with \(x + c\) or \(x - c\), this indicates a horizontal shift.

• A replacement of \(x + c\) shifts the graph to the left by \(c\) units.
• A replacement of \(x - c\) shifts it to the right by \(c\) units.

For example, using our function \(f(x) = x^3\), if you need to shift this graph left by 4 units, you substitute \(x\) with \(x+4\), resulting in \((x+4)^3\).

This transformation alters the position of the entire graph horizontally without affecting its shape. If the original point was \((x, y)\), it will shift to \((x-4, y)\) after the transformation. Remember, moving left involves adding a number inside the function, which might be counterintuitive at first.

Cubic Functions

Cubic functions are polynomial functions of degree three, with the general form \(f(x) = ax^3 + bx^2 + cx + d\). These functions create graphs with specific characteristics, often displaying a curved, elongated S-shape known as a cubic curve.

The simplest cubic function is \(f(x) = x^3\), which passes through the origin and is symmetric with respect to the origin. This graph extends infinitely in both directions.

Cubic functions can exhibit interesting features:

• They can have up to two turning points where the graph changes direction.
• The steepness or width of the curve can vary based on the coefficients \(a, b, c\).
• They may intersect the x-axis up to three times, acting as a real zero of the function.

When you apply any transformations like shifts to a cubic function, you're effectively shifting the entire graph from where these turning points and intercepts originally are. Understanding the fundamental shape of cubic functions allows for better insight into how these transformations will change the function's visual representation on a graph.