Problem 22 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-26: Sketch, on the same coordinate plane, the graphs of \(f\) for the given values of \(c\). (Make use of symmetry, shifting, stretching, compressing, or reflecting.) $$ f(x)=(x+c)^{3} ; \quad c=-2,1,2 $$

Step-by-Step Solution

Answer

Shift the cubic graph horizontally: 2 units right for \(c = -2\), 1 unit left for \(c = 1\), and 2 units left for \(c = 2\).

Step 1: Understand the Base Function

The base function provided is a cubic function, \(f(x) = x^3\). This function is symmetric around the origin and has an inflection point at \(x = 0\). The key transformation to understand is \(f(x + c) = (x + c)^3\), which translates (shifts) the graph horizontally by \(-c\) units.

Step 2: Apply Transformation for c = -2

For \(c = -2\), the function becomes \(f(x) = (x + (-2))^3 = (x - 2)^3\). This shifts the graph of \(x^3\) 2 units to the right. So, plot the cubic graph centered at \(x = 2\).

Step 3: Apply Transformation for c = 1

For \(c = 1\), the function becomes \(f(x) = (x + 1)^3\). This shifts the graph of \(x^3\) 1 unit to the left. So, plot the cubic graph centered at \(x = -1\).

Step 4: Apply Transformation for c = 2

For \(c = 2\), the function becomes \(f(x) = (x + 2)^3\). This shifts the graph of \(x^3\) 2 units to the left. So, plot the cubic graph centered at \(x = -2\).

Step 5: Sketching the Graphs

On the same coordinate plane, draw the three cubic graphs using the transformations identified in steps 2-4. Each graph is similar in shape but is shifted horizontally according to the calculated values of \(c\). Ensure that the characteristic cubic shape and symmetry around their respective new centers are maintained.

Key Concepts

Cubic Functions

Cubic functions are polynomial functions where the highest degree of the variable is three, typically expressed as \( f(x) = ax^3 + bx^2 + cx + d \). These functions feature a characteristic S-shaped curve and are known for their distinct properties, such as having an inflection point, where the curvature changes direction. The simplest form of a cubic function is \( f(x) = x^3 \), which is symmetric about the origin. This type of symmetry is called odd symmetry, meaning that if you rotate the graph 180 degrees around the origin, it looks identical. The cubic function also has no maximum or minimum points, as it extends towards infinity in both the positive and negative directions. As a student, it is important to recognize that the behavior of cubic functions makes them both interesting and crucial for understanding more complex mathematical concepts.

Horizontal Shifts

Horizontal shifts involve moving a graph left or right along the x-axis. For a cubic function, the transformation \( f(x + c) = (x + c)^3 \) effectively moves the graph horizontally by \(-c\) units. Understanding this concept requires recognizing that "+c" inside the function brackets causes a shift in the opposite direction.
For instance:

• When \( c = -2 \), the transformation is \( f(x) = (x - 2)^3 \), which shifts the graph 2 units to the right.
• When \( c = 1 \), we get \( f(x) = (x + 1)^3 \), meaning the graph shifts 1 unit to the left.
• When \( c = 2 \), the equation \( f(x) = (x + 2)^3 \) shifts the graph 2 units to the left.

These shifts are crucial when sketching transformed graphs, as they dictate where on the x-axis the cubic curve will be centered. Practicing horizontal shifts with different values of \( c \) will help you grasp this concept effortlessly.

Function Graphing

Function graphing is the process of plotting a function's output against its input on a coordinate plane. Graphing cubic functions like \( f(x) = (x + c)^3 \) involves recognizing and applying transformations like horizontal shifts.
To accurately graph a cubic function, you should:

• Identify key transformations such as shifts, reflections, and stretches/compressions that impact the graph’s appearance.
• Note the inflection point, which for a basic cubic function remains at the new center after applying any transformations.
• Ensure that the curve maintains its characteristic S-shape.

Additionally, when graphing on the same plane as other transformations of the same function, it’s important to highlight how each graph is similar in shape but differs in position due to the specific transformations applied. This helps in recognizing the relationship between the functions and their respective graphs.

Symmetric Graphs

Symmetry in graphs helps simplify the graphing process by reducing the amount of detail needed to create an accurate sketch. In the case of cubic functions, like \( f(x) = x^3 \), the graph exhibits odd symmetry about the origin. This means that the function satisfies the equation \( f(-x) = -f(x) \). This property is vital as it ensures that the function’s behavior is predictable when transforming and shifting the graph.
For transformations involving horizontal shifts, while the graph's center moves horizontally, the symmetric nature of the cubic curve is maintained around the new center. Understanding this symmetry can help you anticipate a function's appearance after transformations. This reduces the need to plot many individual points and enables you to sketch with confidence, knowing that the curve will balance itself symmetrically once the center is correctly positioned.