Problem 34 - A Graphical Approach to Precalculus with Limits

Question

Give the equation of each function whose graph is described. The graph of \(y=x^{3}\) is vertically stretched by applying a factor of \(3 .\) This graph is then reflected across the \(x\) -axis. Finally, the graph is shifted 8 units upward.

Step-by-Step Solution

Answer

The equation is \(y = -3x^3 + 8\).

Step 1: Start with the Basic Function

The basic function is given by the equation \(y = x^3\). This is the graph that we will modify according to the described transformations.

Step 2: Apply the Vertical Stretch

To apply a vertical stretch by a factor of \(3\), you multiply the entire function by \(3\). So the equation becomes \(y = 3x^3\).

Step 3: Reflect Across the x-axis

Reflecting the graph across the \(x\)-axis involves multiplying the function by \(-1\). After applying this transformation, the function becomes \(y = -3x^3\).

Step 4: Shift the Graph Upward

To shift the graph 8 units upward, we add \(8\) to the function. Therefore, the final transformed function is \(y = -3x^3 + 8\).

Key Concepts

Vertical Stretch

A vertical stretch is a transformation that stretches or compresses the graph of a function vertically. It affects the function's

• height
• and steepness

without altering its width. To perform a vertical stretch, multiply the entire function by a specific factor. For example, when the graph of the function \(y = x^3\) is vertically stretched by a factor of 3, the new function becomes \(y = 3x^3\).

This means that for every given \(x\) value, the corresponding \(y\) value is now three times larger than it was in the original function. It's helpful to visualize this change as the graph appears stretched further away from the x-axis, making it steeper. Note that, if the stretching factor is greater than 1, the graph stretches upwards. If it is between 0 and 1, the graph compresses downward.

Reflection Across the x-axis

Reflecting a function's graph across the x-axis results in a flip over the x-axis, altering the graph's

• orientation

by changing

• all positive y-values to negative
• and negative y-values to positive.

To achieve this reflection, multiply the function by \(-1\). In our exercise, after applying the vertical stretch, we have the function \(y=3x^3\). Reflecting this across the x-axis gives us \(y=-3x^3\).
This transformation maintains the shape of the graph but reverses its direction. Now, what was originally above the x-axis is below it, and vice versa. It's like looking at the graph in a mirror placed along the x-axis. Remember that this doesn't change the function's steepness or width.

Vertical Shift

A vertical shift moves the entire graph of a function up or down along the y-axis. This does not affect

• the shape
• or orientation of the graph

— just its position.

To apply a vertical shift, add or subtract a constant from the function. Adding shifts the graph upward, while subtracting shifts it downward. In the exercise, we shift the function \(y = -3x^3\) 8 units up by adding 8: \(y = -3x^3 + 8\).

This means every point on the graph has moved 8 units up along the y-axis. This transformation affects the overall location of the graph but keeps its shape exactly the same. It's important to note that vertical shifts are always performed after any stretches, compressions, or reflections to ensure the graph's final position is accurate.