Problem 45 - Algebra for College Students

Question

For each of the following, predict the general shape and location of the graph, and then use your calculator to graph the function to check your prediction. (Your knowledge of the graphs of the basic functions that are being added or subtracted should be helpful when you are making your predictions.) (a) \(f(x)=x^{4}+x^{2}\) (b) \(f(x)=x^{3}+x^{2}\) (c) \(f(x)=x^{4}-x^{2}\) (d) \(f(x)=x^{2}-x^{4}\) (e) \(f(x)=x^{2}-x^{3}\) (f) \(f(x)=x^{3}-x^{2}\) (g) \(f(x)=|x|+\sqrt{x}\) (h) \(f(x)=|x|-\sqrt{x}\)

Step-by-Step Solution

Answer

Predict graph shape using base functions; verify predictions with a graphing tool.

Step 1: Understand the Base Functions

Each function is a combination of basic polynomial functions, such as quadratic, cubic, and quartic functions, and some also include an absolute or square root function. Understanding how these basic forms affect the graph is crucial for making predictions about the combined function.

Step 2: Predict Graph of (a)

For the function \(f(x) = x^4 + x^2\), both terms are even functions that create symmetric shapes about the y-axis. The quartic term \(x^4\) dominates for large \(|x|\), giving a 'W' shaped curve. The \(x^2\) term shifts the min points upwards.

Step 3: Predict Graph of (b)

For \(f(x) = x^3 + x^2\), the cubic term \(x^3\) dominates and creates an S-shaped graph passing through the origin. The addition of \(x^2\) will adjust this graph, making it steeper near x = 0.

Step 4: Predict Graph of (c)

In the function \(f(x) = x^4 - x^2\), again an even-function setup exists, but the \(x^2\) is subtracted, flattening the curve around the origin while maintaining symmetry.

Step 5: Predict Graph of (d)

Here \(f(x) = x^2 - x^4\) is similar to (c) but reversed in dominance; it will have a downward-opening 'bowl' shape, inverted due to the dominant \(-x^4\), flattening mainly near zero.

Step 6: Predict Graph of (e)

For \(f(x) = x^2 - x^3\), the cubic term will win out at extremes, the graph passes through (0,0) as a cubic but opens downward due to the negative factor causing a droop.

Step 7: Predict Graph of (f)

For \(f(x) = x^3 - x^2\), the cubic dominance results in an S-shape (through origin) but moderated downwards near zero by the quadratic term.

Step 8: Predict Graph of (g)

In \(f(x) = |x| + \sqrt{x}\), \(|x|\) ensures symmetry and steep slopes from zero, while the \(\sqrt{x}\) (defined for \(x\ge0\)) contributes extra positive values, causing an upward shift and increased leftward curvature.

Step 9: Predict Graph of (h)

For \(f(x) = |x| - \sqrt{x}\), it starts like \(|x|\) but with a downward drag for \(x\ge0\) due to the square root subtraction, leading to higher values for \(x < 0\).

Step 10: Verify with Calculator

Now, use a graphing calculator to plot each function. Visualize them and adjust your predictions as required. Key features should be compared for expected shapes, turns, and trends. Check if each function maintains its predicted characteristics such as symmetry, rise, or fall.

Key Concepts

Polynomial Functions

Polynomial functions are among the most fundamental and versatile functions in mathematics. They are expressions composed of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. A polynomial function can be written as:

• The degree of a polynomial determines its highest power, like in the expression \(x^n\) where \(n\) is the degree.
• The leading term, with the highest power, usually dictates the end behavior of the graph.
• Common polynomial functions include linear (degree 1), quadratic (degree 2), cubic (degree 3), and quartic (degree 4) functions.

Graphs of polynomial functions can vary greatly but usually form continuous, smooth curves without any gaps or jumps. Understanding these basic shapes helps in predicting how complex polynomial combinations will look when plotted.

Even and Odd Functions

In mathematics, exploring whether a function is even or odd can simplify graphing and analysis. Here's how you identify these properties:

• **Even Functions**: These are symmetric about the y-axis. Formally, a function \(f(x)\) is even if \(f(-x) = f(x)\) for all \(x\). For example, \(f(x) = x^2\) is even since \((-x)^2 = x^2\).
• **Odd Functions**: These exhibit symmetry about the origin. For a function to be odd, \(f(-x) = -f(x)\) must hold true. An example is \(f(x) = x^3\), because \((-x)^3 = -(x^3)\).

Recognizing even or odd functions can predict the symmetry in their graphs, greatly aiding in graphing polynomial functions without a calculator.

Graph Symmetry

Graph symmetry focuses on how a graph mirrors itself in different axes, simplifying further graphing predictions and providing deeper insight:

• **Y-axis Symmetry**: If the graph of a function is identical on either side of the y-axis, it has y-axis symmetry, indicating it's an even function.
• **Origin Symmetry**: A graph with origin symmetry typically reflects as an odd function and is mirrored through the point (0,0).
• **X-axis Symmetry**: Rare for function graphs, as this would require multiple y-values for a single x, violating the definition of a function.

Symmetry simplifies the graphing process, helping predict the nature of polynomial graphs and confirming predictions about their shape and behavior.

Function Behavior Prediction

Predicting a function's behavior involves anticipating how it will act over its domain, especially its long-term growth or decline:

• **End Behavior**: This describes how a function behaves as \(x\) approches infinity or negative infinity. For example, in polynomial functions, the leading term's degree and coefficient dictate this behavior.
• **Turning Points**: They represent places where the graph changes direction. A polynomial of degree \(n\) will have at most \(n-1\) turning points.
• **Intercepts**: Where the graph crosses the axes, indicating roots and initial values, crucial for pinning down basic orientation in the coordinate plane.

By leveraging this knowledge, predictions about function shapes, symmetries, and potential graph behaviors can be made even before graphing, offering a valuable tool in mathematical analysis.