Problem 52 - Contemporary Precalculus

Question

Assume \(f(x)=x^{2}-3 .\) Use the standard viewing window to graph the functions \(f\) and \(g\) on the same screen. $$g(x)=f\left(\frac{1}{2} x\right)$$

Step-by-Step Solution

Answer

Question: Describe in words the main differences between the graphs of the functions f(x) = x^2 - 3 and g(x) = f(1/2x). Answer: The main differences between the graphs of the functions f(x) = x^2 - 3 and g(x) = (1/4)x^2 - 3 are that the graph of g(x) is wider than the graph of f(x), and both graphs have the same vertical shift, as they both have -3 as their constant term.

Step 1: Find the expression for g(x)

To find the expression for \(g(x)\), we need to substitute \(\frac{1}{2}x\) into the function \(f(x)\). The given function, \(f(x)=x^2-3\), becomes: $$g(x)=f\left(\frac{1}{2}x\right)=\left(\frac{1}{2}x\right)^2-3$$ Simplify the expression: $$g(x)=\frac{1}{4}x^2-3$$

Step 2: Graph f(x) in the standard viewing window

To graph \(f(x)=x^2-3\), we plot points on the \(x\)-axis from \(-10\) to \(10\). Then determine the corresponding \(y\)-values using the function \(f(x)\). For example, when \(x=-10\), \(f(x)=(-10)^2-3=97\). Repeat this process for all the x-values and sketch the parabola in the standard viewing window.

Step 3: Graph g(x) in the standard viewing window

Similarly for \(g(x)=\frac{1}{4}x^2-3\), plot points on the \(x\)-axis in the range \(-10 \le x \le 10\). Determine the corresponding \(y\) values using the function \(g(x)\). For example, when \(x=-10\), \(g(x)=\frac{1}{4}(-10)^2-3=22\). Repeat this process for all the x-values and sketch the parabola in the standard viewing window along with the plot of \(f(x)\). It will have the same vertical shift as \(f(x)\) but will be wider.

Step 4: Adjust the graph for clarity

To make sure both graphs are easily distinguishable, use different colors or line styles for the two parabolas. Also, don't forget to label each graph with its respective function, i.e., \(f(x)=x^2-3\) and \(g(x)=\frac{1}{4}x^2-3\). With these steps, you should be able to graph both \(f(x)=x^2-3\) and \(g(x)=\frac{1}{4}x^2-3\) in the standard viewing window, on the same screen.

Key Concepts

Standard Viewing Window

When graphing quadratic functions, the standard viewing window refers to a preset range on the graphing calculator or coordinate plane that provides a common baseline for observing the behavior of a graph. In most cases, the standard viewing window for graphing calculators is set to display the x-values from -10 to 10 and the y-values from -10 to 10. This range is often sufficient to view the important features of a quadratic function, such as its vertex and intercepts.

Using the standard viewing window allows students and educators to analyze and compare functions efficiently, as it sets a consistent perspective. However, sometimes, for functions with large or small values, it may be necessary to adjust the viewing window to ensure the graph is properly scaled and all relevant portions are visible.

• For the equation \( f(x) = x^2 - 3 \), setting the x-values from -10 to 10 is efficient for viewing its parabolic shape.
• The graph of \( g(x) = \frac{1}{4}x^2 - 3 \) also fits well within the standard viewing window, though it is wider and requires observing its stretch relative to \( f(x) \).

Being able to use and adjust the standard viewing window is a crucial skill for students to master, as it ensures the accuracy and completeness of their graph analyses.

Transformation of Functions

The concept of transformation of functions is pivotal in algebra and precalculus. A transformation alters an original function's graph in various ways—shifts, stretches, compressions, and reflections. When we apply a transformation to a quadratic function, the parabola may move up or down (vertical shift), left or right (horizontal shift), become wider or narrower (vertical or horizontal stretch/compression), or flip over an axis (reflection).

In the case of the function \( g(x) = f\left(\frac{1}{2}x\right) = \frac{1}{4}x^2 - 3 \), the transformation involves a horizontal stretch by a factor of 2 compared to the parent function \( f(x) \). This happens because the \( x \) values of \( g(x) \) are multiplied by \( \frac{1}{2} \), which dilates the graph of \( f(x) \) horizontally. Therefore, \( g(x) \) appears wider in the standard viewing window than \( f(x) \).

• Understanding transformations helps in predicting the changes in the graph without plotting all the points.
• For visual clarity, different styles or colors can be used to differentiate between the original function and its transformation on the graph.

Recognizing and applying transformations are essential skills in graphing functions, as they provide a foundation for further studies in calculus and higher mathematics.

Parabolas in Precalculus

Parabolas are specific types of curves found in precalculus that represent the graph of a quadratic function of the form \( y = ax^2 + bx + c \). In this equation, a, b, and c are constants with a ≠ 0. The shape and position of a parabola are determined by these constants.

The vertex of the parabola is a significant feature as it represents the maximum or minimum point. For the standard form \( f(x) = x^2 - 3 \), the vertex is at (0, -3), and the parabola opens upward since the coefficient of \( x^2 \) is positive.

• The parabola for \( f(x) \) has its axis of symmetry along the y-axis and opens upwards.
• The transformation that gives us \( g(x) = \frac{1}{4}x^2 - 3 \) doesn't affect the vertex but alters the width of the parabola, making it more spread out along the x-axis.

Understanding the properties of parabolas is vital for students as they delve into topics such as optimization and the analysis of projectile motion in physics. It’s also essential for understanding the effects of different quadratic coefficients on a graph's appearance.