Problem 135 - College Algebra Essentials

Question

a. Use a graphing utility to graph \(f(x)=x^{2}+1\) b. Graph \(f(x)=x^{2}+1, g(x)=f(2 x), h(x)=f(3 x),\) and \(k(x)=f(4 x)\) in the same viewing rectangle. c. Describe the relationship among the graphs of \(f, g, h\), and \(k,\) with emphasis on different values of \(x\) for points on all four graphs that give the same \(y\) -coordinate. d. Generalize by describing the relationship between the graph of \(f\) and the graph of \(g\), where \(g(x)=f(c x)\) for \(c>1\). e. Try out your generalization by sketching the graphs of \(f(c x)\) for \(c=1, c=2, c=3,\) and \(c=4\) for a function of your choice.

Step-by-Step Solution

Answer

The graphs \(f(x)=x^{2}+1, g(x)=f(2x), h(x)=f(3x),\) and \(k(x)=f(4x)\) represent the same parabolic function, but compressed horizontally by a factor of \(c\). The same \(y\)-values occur at points \(x, 2x, 3x,\) and \(4x\) respectively. This is a general feature of the transformation \(f(x) -> f(cx)\) for \(c>1\). When graphing for other functions, the same pattern should be observed.

Step 1: Graph f(x)

Use a graphing utility to graph \(f(x)=x^{2}+1\). You should see a parabola that opens upwards. The vertex is at the origin (0,1).

Step 2: Graph f(x), g(x), h(x), and k(x)

Next, graph \(f(x)=x^{2}+1, g(x)=f(2x), h(x)=f(3x),\) and \(k(x)=f(4x)\) in the same viewing rectangle. Observe and note how the function appears to shrink horizontally as the multiplier on \(x\) increases.

Step 3: Describe Relationships among Graphs

Notice the relationship among the graphs of \(f, g, h,\) and \(k\), specifically looking at different values of \(x\) for points on all four graphs that give the same \(y\)-coordinate. You should find that the same \(y\)-values occur at \(x, 2x, 3x,\) and \(4x\) respectively.

Step 4: Generalize for Any c

Describe the relationship between the graph of \(f\) and the graph of \(g\), where \(g(x)=f(cx)\) for \(c>1\). You should conclude that as \(c\) increases, the graph of \(f(cx)\) appears to shrink toward the y-axis, but the height of the graph remains the same.

Step 5: Test Generalization

Finally, test this generalization by sketching the graphs of \(f(cx)\) for \(c=1, c=2, c=3,\) and \(c=4\) for a new function of your choice. Observe if the same outcomes occur.

Key Concepts

Graphing Utilities

Graphing utilities provide an essential tool for visualizing mathematical concepts, particularly when dealing with quadratic functions. These digital tools take an algebraic expression, like the example provided in our exercise, and transform it into a graphical representation. To illustrate the exercise's quadratic function, enter the formula into the graphing utility as f(x) = x^2 + 1. The resulting graph is a parabola, which is the shape typically associated with quadratic functions.

These utilities offer several advantages. They allow for quick alterations of the function to see how changes affect the graph, such as scaling or shifting the parabola. This immediate visual feedback reinforces understanding of quadratic behavior and its dependence on changes to the coefficients and constants within the equation. Moreover, they can display multiple functions at once, which is incredibly useful in comparing transformations and relationships between different parabolas on the same axes, as in steps b and c of the exercise.

Parabola Transformations

When graphing quadratic functions, transformations provide insights into how changes to an equation affect the parabola's shape and position. In the given exercise, we observe horizontal transformations through the functions g(x) = f(2x), h(x) = f(3x), and k(x) = f(4x).

For each of these, the coefficient c inside the function argument scales the x-values, effectively compressing the graph horizontally towards the y-axis. This type of transformation shows how, for a given y-value, the corresponding x-values are altered by the factor of c. As c increases, the 'squeeze' is more pronounced. This relationship illustrates that parabolas related by such transformations are similar in shape but vary in their width and the spacing between their corresponding points on the x-axis. An important aspect to highlight here is that while the horizontal width changes, the vertical height, as determined by the y-coordinate, remains unchanged. This characteristic is pivotal for fully grasping the impact of horizontal scaling on parabolas.

Quadratic Function Characteristics

Understanding the characteristics of a quadratic function is fundamental in deciphering its graphical representation. The quadratic function, generally denoted as f(x) = ax^2 + bx + c, where a, b, and c are constants, has several key features. Firstly, the graph of a quadratic function is a parabola. If a > 0, the parabola opens upwards, and if a < 0, it opens downwards.

The vertex of the parabola is a vital point, indicating the maximum or minimum value of the function. In the exercise, the vertex is (0,1), indicating that the minimum value of y is 1 when x is 0. The axis of symmetry, another significant feature, is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. In the given function, the axis of symmetry is the y-axis itself. Lastly, when analyzing the function characteristics, consider the intercepts—the points where the parabola intersects the axes. Here, the y-intercept is at (0,1), as the parabola touches the y-axis at y = 1 and does not intersect the x-axis because the function has no real roots.