Problem 1 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

Shift the cubic curve up 3 units for \(c=3\) and down 3 units for \(c=-3\).

Step 1: Identify the Base Function

The base function provided is a cubic function given by \( f(x) = 2x^3 + c \). It is a standard cubic function shifted vertically by \(c\). We'll use \(c = 3\) for part (a) and \(c = -3\) for part (b).

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

The constant \(c\) affects the vertical position of the graph. For \( c = 3 \), the entire graph of \( f(x) = 2x^3 \) is shifted up by 3 units. Similarly, for \( c = -3 \), the graph is shifted down by 3 units.

Step 3: Sketch the Graph for c=3

For part (a), the function becomes \( f(x) = 2x^3 + 3 \). Start by sketching the graph of \( g(x) = 2x^3 \), which passes through the origin and has a symmetric cubic shape. Then, move the entire graph up by 3 units. This results in the graph passing through the point (0,3) and maintaining its cubic shape.

Step 4: Sketch the Graph for c=-3

For part (b), the function becomes \( f(x) = 2x^3 - 3 \). Again, sketch the graph of \( g(x) = 2x^3 \) first. Now, shift this graph down by 3 units. This new graph will pass through the point (0, -3) while preserving the cubic curve shape.

Key Concepts

Graph Transformations

Cubic functions, like the function given by \( f(x) = 2x^3 + c \), can undergo various transformations to alter their appearance on a graph. One common transformation is a shift, which can affect the graph's position without changing its shape.
A shift based on the constant \( c \) is known as a vertical shift, which we'll discuss in detail soon. But graph transformations can also include:

• Horizontal shifts, which move the graph left or right.
• Reflections, which flip the graph across an axis.
• Stretching or compressing, which alters the graph's width or height.

Understanding these transformations helps in sketching and predicting function behavior. They allow us to take a basic graph and modify its position or orientation according to specified parameters.

Vertical Shifts

Vertical shifts specifically refer to moving the entire graph of a function up or down by a certain number of units. This is done without altering the shape of the graph itself.
In our cubic function \( f(x) = 2x^3 + c \), the term \( c \) directly influences the vertical shift:

• If \( c > 0 \), the graph of the function shifts up by \( c \) units.
• If \( c < 0 \), the graph shifts down by \( c \) units.
• When \( c = 0 \), the function remains in its original position, passing through the origin.

For example, with \( c = 3 \), the function \( f(x) = 2x^3 + 3 \) moves every point on the original graph \( 2x^3 \) up by 3 units. This results in the graph passing through \( (0, 3) \).
Conversely, with \( c = -3 \), the function \( f(x) = 2x^3 - 3 \) shifts the graph down by 3 units, aligning through \( (0, -3) \). These simple shifts are critical in understanding how graphs can be manipulated while maintaining their core structure.

Function Analysis

Analyzing functions like \( f(x) = 2x^3 + c \) involves examining their behavior and characteristics. Understanding a cubic function's movement due to transformations like vertical shifts gives insights into its graph.
Cubic functions inherently have a symmetric S-shaped curve due to their degree. This function opens upwards or downwards depending on the sign of the leading coefficient:

• If the coefficient is positive, the arms of the graph extend to \( \, \infty \).
• With a negative coefficient, it extends towards \( -\infty \).

Vertical shifts simply move this entire curve up or down without changing its fundamental properties. Analyzing where the graph intersects the y-axis, especially after a shift, helps to map out precise graph positions.
By observing these transformations, it becomes possible to make predictions about a function's graph, behavior, and its intersection with axes uniquely for each value of \( c \). Example analyses like these help uncover the relationship between algebraic equations and their graphical representations, making concept comprehension much more intuitive.