Problem 38 - Algebra for College Students

Question

Graph \(f(x)=x^{4}+x^{3}\). Now predict the graph for each of the following, and check each prediction with your graphing calculator. (a) \(f(x)=x^{4}+x^{3}-4\) (b) \(f(x)=(x-3)^{4}+(x-3)^{3}\) (c) \(f(x)=-x^{4}-x^{3}\) (d) \(f(x)=x^{4}-x^{3}\)

Step-by-Step Solution

Answer

Graphs are shifted, reflected, or altered by each operation; check graphs with a calculator.

Step 1: Understand the Base Function

Graph the base function \( f(x) = x^4 + x^3 \). This function is a polynomial of degree 4 with no shifts and can be sketched by testing and plotting a few key x-values and their corresponding y-values, and then sketching the curve based on the behavior of the function as \( x \to \pm \infty \).

Step 2: Predict the Effect of Constant Shift

For \( f(x) = x^4 + x^3 - 4 \), predict that the graph of \( f(x) = x^4 + x^3 \) will shift downward by 4 units, since subtracting 4 affects only the vertical position.

Step 3: Predict the Horizontal Shift

For \( f(x) = (x-3)^4 + (x-3)^3 \), predict the graph will shift 3 units to the right compared to the graph of \( f(x) = x^4 + x^3 \), because replacing \( x \) with \( x-3 \) shifts the function horizontally to the right.

Step 4: Predict the Reflection

For \( f(x) = -x^4 - x^3 \), predict the graph will be a reflection across the x-axis of the graph of the original function \( f(x) = x^4 + x^3 \). This inversion occurs because multiplying by -1 reflects the graph over the x-axis.

Step 5: Predict the Change in Sign of Cubic Term

For \( f(x) = x^4 - x^3 \), predict that the graph will retain the general shape of \( x^4 \) in the end behavior, but the presence of \(-x^3\) will alter the symmetry and turning points of the graph near the origin. Expect this to affect the mid-section curvature compared to \( f(x) = x^4 + x^3 \).

Step 6: Check Predictions with a Graphing Calculator

Use a graphing calculator to input each modified function and compare the resulting graphs to ensure the predictions align with the actual graphs. Evaluate how constants, reflections, and shifts visually alter the base graph.

Key Concepts

Graph Transformations

Graph transformations help us understand how a function can change in shape or position. For polynomial functions like \(f(x) = x^4 + x^3\), transformations can arise from:

• Vertical shifts
• Horizontal shifts
• Reflections
• Stretching or compressing

To grasp these changes, imagine the base graph as a starting point. Any transformation comes as an adjustment to this base, altering either its position or shape. Transformations are crucial for predicting how graphs of modified functions will look, based on adjustments from the base function.

Polynomial Graphs

Polynomial graphs, such as those created by functions like \(x^4 + x^3\), display varied and interesting shapes. Here's what you should know:

• They are smooth and continuous curves.
• The degree of the polynomial tells us the most extreme behavior at either end of the graph. For our base function, the degree is 4.
• The degree also indicates the maximum number of turning points, which for a polynomial of degree \(n\), can be at most \(n-1\).

When graphing polynomials, test and plot key points to visualize the curve. Recognize how these graphs change with transformations to better predict their behavior before using a graphing calculator.

Reflection of Graphs

Reflections are a pivotal concept in graph transformations, especially in polynomial functions. A reflection flips a graph across a line, often the x-axis.For example, considering \(f(x) = -x^4 - x^3\), reflection across the x-axis happens by multiplying the function by \(-1\). This changes every positive y-value to a negative one, and vice versa. So, the entire graph inverts. Reflections help us understand how the sign of the function's terms influences the graph's orientation.

Graph Shifts

Graph shifts can be vertical or horizontal, altering the position of the graph without changing its shape. Let's explore each type:Vertical shifts adjust the graph up or down. Subtracting 4 in \(f(x) = x^4 + x^3 - 4\) shifts the base graph downward by 4 units. Vertical shifts affect the y-coordinate directly.

Horizontal shifts take place when the variable \(x\) is replaced by \(x - b\). In \(f(x) = (x-3)^4 + (x-3)^3\), the replacement shifts the graph 3 units to the right. This happens because the x-coordinate of every point on the graph increases by 3.

End Behavior of Functions

End behavior describes how the outputs of a function behave as the inputs head to positive or negative infinity. It's especially relevant for polynomial functions.For \(f(x) = x^4 + x^3\), as \(x \to \pm \infty\), the function's degree tells us that the ends of the graph will rise to \(+\infty\). In this case, since 4 is an even number, both sides move upwards.

Understanding end behavior is crucial in predicting the general shape of the graph and deciphering how it might be transformed by reflecting or altering polynomial terms, like seeing how \(-x^3\) influences the original graph's curvature.