Problem 52 - 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, \(x \geq 0\) $$f(x)=x^{1.5}$$

Step-by-Step Solution

Answer

Graph \(f(x) = x^{1.5}\) and \(y = x^2\). \(f(x)\) grows slower for \(x > 1\).

Step 1: Identify and Compare Functions

We need to sketch two functions on the same graph: the given power function \(f(x) = x^{1.5}\) and the quadratic function \(y = x^2\). First, understand that \(f(x) = x^{1.5}\) is a power function with an exponent between 1 and 2. This function will generally lie between the linear and quadratic functions for our range \(x \geq 0\).

Step 2: Choose Points for Evaluation

Choose several non-negative values of \(x\) to evaluate both functions. It's useful to choose values like \(x = 0, 1, 2, 3, 4\) to see the behavior at low, moderate, and higher values of \(x\).

Step 3: Calculate \(y\)-Values for \(f(x) = x^{1.5}\)

Calculate the values of \(f(x) = x^{1.5}\) for the selected values of \(x\). - \(f(0) = 0^{1.5} = 0\)- \(f(1) = 1^{1.5} = 1\)- \(f(2) = 2^{1.5} \approx 2.83\)- \(f(3) = 3^{1.5} \approx 5.2\)- \(f(4) = 4^{1.5} = 8\).

Step 4: Calculate \(y\)-Values for \(y = x^2\)

Now, calculate the values of \(y = x^2\) for the same points:- \(y(0) = 0^2 = 0\)- \(y(1) = 1^2 = 1\)- \(y(2) = 2^2 = 4\)- \(y(3) = 3^2 = 9\)- \(y(4) = 4^2 = 16\).

Step 5: Plotting and Sketching the Graphs

On a graph with \(x\)-axis representing the input values and \(y\)-axis representing the outputs, plot points for both functions. Make sure to properly label each curve as \(f(x) = x^{1.5}\) and \(y = x^2\). Compare the shapes; \(f(x) = x^{1.5}\) should be positive and increasing, but grow slower than \(x^2\) for \(x > 1\).

Step 6: Final Graphical Comparison

Upon completion of plotting, you should see two distinct curves. The curve for \(x^{1.5}\) will start below \(x^2\) for \(x > 1\) but will share points and come close near \(x = 1\). Verify this as it will give insight into the slower acceleration of \(y = x^{1.5}\) when compared with \(y = x^2\).

Key Concepts

Graphing Functions

Graphing functions is a crucial skill in mathematics, helping us visualize relationships and understand how one variable affects another. By plotting a function on a graph, we can easily see how the values change and make connections between different functions.
When graphing, the x-axis typically represents our input values, while the y-axis shows the output of our function. In this exercise, we're focusing on the power function \( f(x) = x^{1.5} \) and the quadratic function \( y = x^2 \).

• Start by choosing several values for \( x \), such as \( 0, 1, 2, 3, \) and \( 4 \).
• Calculate the corresponding \( y \)-values for each function and plot these points on the graph.
• Connect the points to form smooth, continuous curves for each function.

This will give you a visual representation of each function's behavior, where you can compare them directly.When comparing, observe that for \( x \geq 0 \), \( f(x) = x^{1.5} \) lies between the linear function and \( y = x^2 \), with a growth rate faster than linear but slower than quadratic.

Exponents

Exponents are fundamental to understanding power functions. They represent how many times a number, called the base, is multiplied by itself. In the function \( f(x) = x^{1.5} \), the exponent \( 1.5 \) tells us how \( x \) is being transformed.
An exponent like \( 1.5 \) is considered a fractional exponent, which can be broken down into more known roots and powers. It can be interpreted as \( x^{1.5} = x^{1 + 0.5} = x^1 \times x^{0.5} \), signifying the square root of \( x \) times \( x \) itself.

• A base raised to a power greater than 1, like \( x^{1.5} \), results in a graph that increases and slopes upwards.
• Fractional exponents like \( 1.5 \) result in a more moderate increase than whole number exponents.

Understanding these basic rules of exponents helps effectively manipulate and graph functions with different power components. Fractional powers like \( 1.5 \) often provide unique insights into function growth and behavior.

Quadratic Functions

Quadratic functions are a central theme in algebra, characterized by their standard form \( y = ax^2 + bx + c \) and their distinctive parabolic shape. In this exercise, we dealt with the basic quadratic \( y = x^2 \).
Key features of quadratic functions include:

• They graph as a parabola that is symmetric around its vertex, typically a minimum or maximum point.
• The power of 2 results in the function growing quickly as \( x \) moves away from the origin.
• For \( y = x^2 \), the vertex is at the origin \((0, 0)\) and it opens upwards.

Quadratics like \( y = x^2 \) help us understand growth that accelerates over time, contrasting with linear or slower growers like \( y = x^{1.5} \). Comparing \( x^{1.5} \) to \( x^2 \), the latter shows faster acceleration for \( x > 1 \), clearly seen in their plotted curves, which diverge as \( x \) gets larger. Recognizing these distinctions helps in the broader study of polynomials and their applications.