Problem 36 - Intermediate Algebra

Question

(a) Graph \(y=x^{2}, y=2 x^{2}, y=3 x^{2}\), and \(y=4 x^{2}\) on the same set of axes. (b) Graph \(y=x^{2}, y=\frac{3}{4} x^{2}, y=\frac{1}{2} x^{2}\), and \(y=\frac{1}{5} x^{2}\) on the same set of axes. (c) Graph \(y=x^{2}, y=-x^{2}, y=-3 x^{2}\), and \(y=-\frac{1}{4} x^{2}\) on the same set of axes.

Step-by-Step Solution

Answer

Graph different quadratic functions on the same set of axes as described: wider parabolas have smaller 'a' values, and negative 'a' values point downwards.

Step 1: Understanding the Problem

We are given three separate parts to solve: (a), (b), and (c). Each part requires us to graph specific quadratic functions on the same set of axes. Quadratic functions generally have the form of a parabola, and the 'a' coefficient in front of the \(x^2\) term determines the width and direction.

Step 2: Graphing Part (a)

For part (a), we need to graph \(y = x^{2}, y = 2x^{2}, y = 3x^{2},\) and \(y = 4x^{2}\). All these are upward-opening parabolas centered at the origin. The larger the coefficient in front of \(x^2\), the narrower the parabola. Start by drawing these parabolas, noting that \(y = x^{2}\) is the widest, \(y = 4x^{2}\) is the narrowest.

Step 3: Graphing Part (b)

For part (b), we graph \(y = x^{2}, y = \frac{3}{4}x^{2}, y = \frac{1}{2}x^{2},\) and \(y = \frac{1}{5}x^{2}\). These functions also open upwards. Here, \(y = x^{2}\) is narrowest, and \(y = \frac{1}{5}x^{2}\) is the widest. Plot these on the same set of axes as well, noting how the decreasing 'a' values make the parabolas wider.

Step 4: Graphing Part (c)

For part (c), graph \(y = x^{2}, y = -x^{2}, y = -3x^{2},\) and \(y = -\frac{1}{4}x^{2}\). \(y = x^{2}\) opens upwards, while the others open downwards because of the negative coefficients. \(-x^{2}\) is the standard downward parabola, \(-3x^{2}\) is narrower, and \(-\frac{1}{4}x^{2}\) is wider due to the smaller magnitude of the coefficient.

Key Concepts

Parabolas

When we talk about parabolas in the context of quadratic functions, we're referring to the U-shaped curves that these functions create on a graph. Each quadratic function produces a unique parabola based on its specific equation. These parabolas can open upwards or downwards, depending on the coefficients in the equation.What makes a parabola unique? It's primarily the coefficient of the squared term, usually represented as 'a' in the quadratic function format of \(y = ax^2 + bx + c\). This 'a' coefficient affects the direction and the curvature of the parabola. When 'a' is positive, the parabola opens upwards like a smile. When 'a' is negative, it opens downwards like a frown. These changes in direction and shape are crucial in understanding how different quadratic equations behave visually on a graph.

Graphing Quadratic Equations

Graphing quadratic equations involves plotting points on a coordinate grid to represent the solutions of the equation. For a quadratic equation of the form \(y = ax^2 + bx + c\), the graph will always be a parabola. The task of graphing these equations helps us visually comprehend how the solutions to the equations map out as a continuous curve.Here's a simple way to approach graphing:

• Start by identifying the vertex, the highest or lowest point of the parabola, which can be found using the formula for the axis of symmetry.
• Next, calculate key points by substituting chosen x-values into the equation to find corresponding y-values.
• Use the symmetry of the parabola about the axis of symmetry to plot points on both sides, facilitating a balanced curve.

This process translates the quadratic equation into a visual form, providing deeper insights into its properties and behavior.

Coefficient Effects

The coefficient 'a' in the quadratic equation \(y = ax^2 + bx + c\) plays a significant role in shaping the parabola. Changes in this coefficient can make the parabola appear wider or narrower and also determines its direction relative to the x-axis.

• When the absolute value of 'a' is large, the parabola becomes narrower. This is because the rate at which the y-values increase or decrease is more pronounced.
• Conversely, when 'a' has a small absolute value, the parabola is wider. This means that the curve extends gently on both sides of the vertex.
• A positive 'a' means the parabola opens upwards, creating a U-shape, whereas a negative 'a' flips it downwards into an inverted U.

Learning the effect of the coefficient helps us adjust and predict the changes in a parabola's shape and direction based on different quadratic equations.

Axis of Symmetry

The axis of symmetry is a essential feature of any parabola, providing a vertical line that divides the parabola into two mirror-image halves. For any quadratic equation in the form \(y = ax^2 + bx + c\), the axis of symmetry can be calculated using the formula \(x = -\frac{b}{2a}\).This axis not only helps in plotting the parabola accurately but also in identifying key properties such as the vertex:

• Locating the vertex can be simplified once the axis of symmetry is known, as the x-value of the vertex lies on this line.
• The symmetry about this axis means that if you know one half of your parabola, you effectively know the other half as well.

Grasping the concept of the axis of symmetry is invaluable when analyzing and graphing quadratic equations, offering a straightforward pathway to understanding the parabola's shape and position on the graph.