Problem 4 - Precalculus: Functions and Graphs

Question

Sketch the graph of \(f\) for the indicated value of \(c\) or \(a\) $$f(x)=a x^{3}-3$$ (a) \(a=-2\) b) \(a=\frac{1}{4}\)

Step-by-Step Solution

Answer

For \( a = -2 \), the graph is reflected and steeply decreases. For \(\frac{1}{4}\), it gradually increases with a vertical compression.

Step 1: Understand the Function

The function given is \( f(x) = a x^3 - 3 \), where \( a \) is a parameter that affects the steepness and direction of the graph. The term \(-3\) shifts the entire graph vertically downwards by 3 units.

Step 2: Analyze the Effect of 'a' on the Graph

The value of \( a \) determines whether the cubic graph opens upwards or downwards. If \( a > 0 \), the graph opens upwards from the left to the right. If \( a < 0 \), the graph opens downwards. Additionally, different values of \( a \) change the steepness of the curve.

Step 3: Graph f(x) for a = -2

For \( a = -2 \), substitute into the function to get \( f(x) = -2x^3 - 3 \). This means the graph of \( x^3 \) is reflected over the x-axis (because \( a \) is negative), vertically stretched (because the absolute value of \( a \) is more than 1), and shifted downward by 3 units. The graph will decrease from the left, flatten at the origin, and continue decreasing.

Step 4: Graph f(x) for a = \(\frac{1}{4}\)

For \( a = \frac{1}{4} \), substitute into the function to get \( f(x) = \frac{1}{4}x^3 - 3 \). This means the graph of \( x^3 \) is compressed vertically (because the absolute value of \( a \) is less than 1) and shifted downward by 3 units. The graph will increase gradually from the left, flatten at the origin, and continue increasing to the right.

Step 5: Sketching the Graphs

Now, sketch the graphs based on the previous steps:- For \( a = -2 \): Start at the origin, reflect the cubic graph across the x-axis, apply a vertical stretch, and shift down by 3 units.- For \( a = \frac{1}{4} \): Start at the origin, apply a vertical compression, and shift down by 3 units. Draw the gradually increasing path of the graph.

Key Concepts

Vertical Shifts

In cubic functions like \( f(x) = a x^3 - 3 \), understanding vertical shifts is crucial for sketching the graph accurately. The term \(-3\) in the equation indicates a vertical shift. This means that every point on the basic graph of \( x^3 \) is moved down by 3 units.

Think of it as sliding the entire graph downwards without changing its shape. Vertical shifts do not affect the symmetry or any aspect of the base graph's orientation. Thus, if you plot the graph on a coordinate plane, instead of beginning at the origin \(0, 0\), the graph moves to \(0, -3\).

Vertical shifts are straightforward transformations, acting independently from any coefficients of \(x^3\) found in the function.

Graph Transformations

Graph transformations modify the basic shape and position of the graph of a function. They include operations such as vertical and horizontal shifts, reflections, and changes in steepness or orientation.

For example, in the function \( f(x) = a x^3 - 3 \), several transformations are taking place:

• The term \(-3\) results in a vertical shift as discussed earlier.
• The parameter \(a\) plays a critical role, further impacting the graph by altering its steepness and direction.

Graph transformations help us understand complex behaviors a function can exhibit. These transformations allow us to manipulate and predict graph shapes without plotting numerous points.

Reflections

Reflections in cubic functions occur when the sign of \(a\) in \( f(x) = a x^3 - 3 \) is negative. This results in flipping the graph over the x-axis. When \( a = -2 \), the graph behaves as a mirror image of the \( x^3 \) graph that would occur if \( a \) were positive.

Reflection across the x-axis does not change the absolute steepness but changes how the graph approaches positive and negative directions. For \( a = -2 \), where the graph of \( ax^3 \) initially increases, it will now decrease because of this reflection.

Think of reflection as altering the paths and endpoints of the graph—it will flip any increasing segments to decreasing ones, creating an inverse scenario.

Stretching and Compression

Stretching and compression modify the steepness of the graph. In our function, \( f(x) = a x^3 - 3 \), the absolute value of \(a\) dictates this transformation.

If \(|a| > 1\), the graph will stretch vertically, making it steeper. This occurs with \(a = -2\) in our example, creating a steeper graph that has been vertically elongated.

Conversely, if \(|a| < 1\), the graph compresses vertically, resulting in a flatter appearance. This happens with \(a = \frac{1}{4}\), where the graph becomes less steep, showcasing a gradual increase.

Vertical stretching and compression play an important role in defining how rapidly the function values change as \(x\) moves away from zero. They reflect the function's sensitivity to variations in the input \(x\).