Problem 21 - Thomas Calculus

Question

Exercises \(19-28\) tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation. $$ y=x^{3} \quad \text { Left } 1, \text { down } 1 $$

Step-by-Step Solution

Answer

Shifted equation: \(y = (x+1)^3 - 1\).

Step 1: Identify Shift Directions

The problem states that the graph of the equation \(y=x^3\) needs to be shifted left by 1 unit and down by 1 unit.

Step 2: Horizontal Shift Adjustment

To shift the graph to the left by 1 unit, we replace \(x\) with \(x+1\). This results in the intermediate equation \(y = (x+1)^3\).

Step 3: Vertical Shift Adjustment

To shift the graph down by 1 unit, we subtract 1 from the entire equation. Applying this to the intermediate equation, we get \(y = (x+1)^3 - 1\).

Step 4: Write Down the Final Shifted Equation

Combining both shifts, the shifted equation is \(y = (x+1)^3 - 1\). This is the equation of the graph after applying the horizontal and vertical shifts.

Step 5: Graph Sketching

Sketch the original graph of \(y = x^3\), which is a typical cubic curve passing through the origin. Next, shift all the points 1 unit to the left and 1 unit down to get the shifted graph of \(y = (x+1)^3 - 1\). Label each graph with its respective equation.

Key Concepts

Horizontal Shift

When we talk about a horizontal shift in a graph, it's all about moving the entire graph either to the left or to the right along the x-axis. To perform a horizontal shift on a function, you need to adjust the function's input, or x-value.
For example, if you want to move a graph to the left by 1 unit, you would replace each occurrence of x in your function with \((x+1)\). This might seem counterintuitive at first because to move left, we add. It's important to remember:

• To shift right, replace \(x\) with \(x - a\).
• To shift left, replace \(x\) with \(x + a\).

In our exercise, we started with the function \(y = x^3\) and moved it left by 1 unit, transforming it into \(y = (x+1)^3\). This means every point is taken 1 unit to the left on the graph.

Vertical Shift

Vertical shifts are another essential transformation where the entire graph is moved up or down along the y-axis. Unlike a horizontal shift, a vertical shift changes the function's output.
This is much more intuitive:

• To shift up, you add \(a\) to the entire function.
• To shift down, you subtract \(a\) from the entire function.

In our exercise, we needed to shift the graph down by 1 unit. So, after applying the horizontal shift and getting \(y = (x+1)^3\), we subtracted 1 from the whole function, giving us \(y = (x+1)^3 - 1\). This ensures every point moves 1 unit down on the graph.

Cubic Functions

Cubic functions are a type of polynomial function characterized by the highest exponent being three. The general form is \(y = ax^3 + bx^2 + cx + d\). They are known for their distinctive S-shaped curves, having one or two bends.
The function \(y = x^3\) is the simplest cubic function. It's symmetric around the origin and passes through it, creating a curve that moves upwards to the right and downwards to the left.
This exercise considers transformations on this basic form. In transformations like horizontal and vertical shifts, the overall shape of the cubic curve does not change; only its position does. This neat feature of cubic functions makes them very predictable when applying transformations. By shifting \(y = x^3\) to \(y = (x+1)^3 - 1\), we alter the position but preserve the S-curve's unique character.