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Magazine Chapter 11: Problem 58
Graph the given functions. $$f(x)=2 x^{2 / 3}$$
Short Answer
Expert verified
Graph \( f(x) = 2 x^{2/3} \) has symmetry about the y-axis and forms a curve similar to an opening 'V' centered at the origin.
Step by step solution
01
Identify the Type of Function
The function given is \( f(x) = 2x^{2/3} \). This function is a transformation of the parent function \( x^{2/3} \), which is a power function. It will have symmetry and specific behaviors we will explore graphically.
02
Analyze the Function's Structure
The exponent \( \frac{2}{3} \) indicates a root and a power. The graph of \( y = x^{2/3} \) takes a square root for every cubed input, creating a more gradual increase compared to parabolas. This function will be positive for all real \( x \), as the input is raised to the power of \( 2 \).
03
Determine Key Features
The function passes through the origin (0,0), as substituting \( x = 0 \) gives \( f(0) = 0 \). It is symmetric about the y-axis because the even exponent affects both positive and negative values equally. As \( x \) increases positively or negatively, \( f(x) \) will increase.
04
Plot Critical Points
Choose several points to plot the function: \( f(-1) = 2(-1)^{2/3} = 2 \); \( f(0) = 0 \); \( f(1) = 2(1)^{2/3} = 2 \). These points show the symmetry and confirm the function's main features. Plot these on a coordinate grid.
05
Sketch the Graph
Using the points from Step 4, sketch the curve on a graph. The graph should appear similar to a curved 'V', originating at the origin and opening upwards and outwards, due to the square root and cubed properties balancing as x values grow.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Power Functions
Power functions are a delightful part of mathematics. They are expressions of the form \( f(x) = ax^n \). Here, \( a \) is a coefficient, and \( n \) is a real number. The way a power function behaves depends largely on the value of \( n \).
- If \( n \) is a positive integer, the graph will typically pass through the origin. The larger the value of \( n \), the steeper the curve.
- If \( n \) is a fraction, like in our example \( f(x) = 2x^{2/3} \), the function could involve roots, resulting in smoother transitions on the graph.
- Negative exponents indicate reciprocal functions, which can be a bit trickier and introduce asymptotes.
Symmetry in Functions
Symmetry in graphs makes them easier to understand and predict. Symmetrical graphs exhibit balanced and mirrored behaviors across an axis.
- Even functions, such as \( f(x) = x^2 \), show symmetry about the y-axis, meaning if you fold the graph over this axis, both halves would match.
- Odd functions, like \( f(x) = x^3 \), instead show symmetry about the origin itself. This is like rotating the graph 180 degrees around the origin and finding it looks the same.
- The function \( f(x) = 2x^{2/3} \) is symmetric about the y-axis because the exponent \( 2 \) in \( 2/3 \) is even, affecting positive and negative x equally.
Critical Points
Critical points are essential in understanding how functions behave. These points inform us about aspects like where the function changes direction or has peaks and valleys.
- Critical points occur where the function's derivative is zero or undefined.
- In \( f(x) = 2x^{2/3} \), a critical point occurs at the origin (0,0). At this point, the derivative does not exist due to the root nature in the exponent \( 2/3 \).
- Besides (0,0), selected points like \(-1\) and \(1\) also highlight symmetry and function characteristics, where \( f(-1) = f(1) = 2 \).
Coordinate Grid Plotting
Plotting functions on a coordinate grid is a fundamental step in visualizing them. This involves setting specific points on a grid representing the input-output relationship.
- Selecting points involves choosing a range of x-values, calculating their corresponding y-values, and marking them on the grid.
- For \( f(x) = 2x^{2/3} \), calculate values such as \( f(-1) \), \( f(0) \), and \( f(1) \): this results in values of \( (鈭�1, 2), (0, 0), (1, 2) \).
- These points reveal the curve's direction and growth, allowing a smooth sketch between them that reflects general behavior.
