Problem 51 - A Graphical Approach to College Algebra

Question

Sketch the graph of each power function by hand, using a calculator only to evaluate \(y\)-values for your chosen \(x\) -values. On the same axes, graph \(y=x^{2}\) for comparison. In each case, \(x \geq 0\) $$f(x)=x^{1.05}$$

Step-by-Step Solution

Answer

Graph \(f(x) = x^{1.05}\) is a curve between \(y = x\) and \(y = x^2\) on the same axes.

Step 1: Understanding Power Functions

The given function is a power function of the form \(f(x) = x^{1.05}\). This function is slightly above the linear function \(f(x) = x\) and slightly below the function \(f(x) = x^2\). The comparison function given is \(y = x^2\). We'll compare \(f(x) = x^{1.05}\) and \(y = x^2\) to understand their shapes and behaviors.

Step 2: Select Sample x-values

Choose a set of \(x\)-values to sketch the graph. Since \(x \geq 0\), common sample points are \(x = 0, 1, 2, 3, 4\). These values will be used to evaluate and compare \(f(x) = x^{1.05}\) and \(y = x^2\).

Step 3: Calculate y-values for f(x) and y = x^2

Using a calculator, evaluate \(f(x) = x^{1.05}\) for each chosen \(x\)-value: - For \(x = 0\), \(f(0) = 0\) - For \(x = 1\), \(f(1) = 1^{1.05} = 1\) - For \(x = 2\), \(f(2) \approx 2.07\) - For \(x = 3\), \(f(3) \approx 3.16\) - For \(x = 4\), \(f(4) \approx 4.22\)Now calculate \(y = x^2\):- For \(x = 0\), \(y = 0\) - For \(x = 1\), \(y = 1\) - For \(x = 2\), \(y = 4\) - For \(x = 3\), \(y = 9\) - For \(x = 4\), \(y = 16\)

Step 4: Plot Points and Compare Graphs

Plot the points \((x, f(x))\) on a graph: \((0, 0), (1, 1), (2, 2.07), (3, 3.16), (4, 4.22)\). On the same axes, plot the function \(y = x^2\) using the points: \((0, 0), (1, 1), (2, 4), (3, 9), (4, 16)\).

Step 5: Sketch the Graph of f(x) = x^{1.05}

Draw a smooth curve through the points for \(f(x) = x^{1.05}\). The curve will be slightly above the line \(y = x\) and below the parabola \(y = x^2\), due to the exponent being just above one.

Step 6: Analyze and Compare Graphs

The graph of \(f(x) = x^{1.05}\) is a gradual curve that rises more sharply than a straight line but more slowly than \(y = x^2\). It provides a smoother slope than \(y = x^2\) due to the exponent being close to 1. Above the line \(y = x\), it starts to diverge more evidently as \(x\) increases.

Key Concepts

Graphing Power Functions

Graphing power functions helps us visualize the growth behavior of specific mathematical expressions. A power function takes the form of \( f(x) = x^n \), where \( n \) can be any real number.In our exercise, we have the function \( f(x) = x^{1.05} \). This particular function means we're raising \( x \) to the power of 1.05, which slightly increases the curve's steepness over a simple linear function, like \( y = x \). To graph this function by hand:

• First, choose several positive \( x \)-values that you can compute easily, such as 0, 1, 2, 3, and 4.
• Then calculate each corresponding \( y \)-value using the formula \( y = x^{1.05} \).
• Plot these points on a coordinate grid.
• Once done, sketch a smooth curve that connects these points.

The result is a smooth curve that rises more steeply than the line \( y = x \), though not as sharply as \( y = x^2 \). It showcases the power of being more than 1 but less than 2.

Comparing Functions

Comparing different types of functions allows us to understand their varying rates of change. In our case, we are comparing \( f(x) = x^{1.05} \) with \( y = x^2 \). These functions are both considered power functions but with different exponents. Here's what to observe:

• **Growth Rate**: The main difference between these functions lies in how quickly their output values grow as \( x \) increases.
• For \( f(x) = x^{1.05} \), the increase is subtle because the power is just a bit above one.
• In contrast, \( y = x^2 \) grows much faster since its exponent is 2.

When plotted together, \( f(x) = x^{1.05} \) will appear below \( y = x^2 \) for all positive \( x \)-values. At small values of \( x \), they may look quite similar, but as \( x \) increases, \( y = x^2 \) will clearly diverge upwards more rapidly than \( f(x) = x^{1.05} \).This comparison shows how even small changes in exponents can lead to significant differences in function behavior.

Calculating Y-Values

To graph functions effectively, calculating \( y \)-values for chosen \( x \)-values is essential. Here's how you can find the \( y \)-values for a power function like \( f(x) = x^{1.05} \).

• Select various \( x \)-values, such as 0, 1, 2, 3, and 4, to cover a range of points.
• Use a calculator to compute each \( y \)-value by substituting the \( x \)-values into the function.
  • For example, \( f(2) \approx 2^{1.05} \approx 2.07 \)
  • Repeat this process for each selected \( x \)-value.

Doing this manually helps in appreciating how the small decimal in the exponent influences the curve shape. Once calculated, these \( y \)-values provide necessary points for plotting the graph accurately. They also serve as a basic tool in comparing the output behavior of different functions, such as in our example with \( y = x^2 \). This hands-on approach enhances the understanding of the function's growth pattern.