Problem 24 - Algebra 2 and Trigonometry

Question

a. Sketch the graph of \(y=x^{2}\) b. Sketch the graph of \(y=3 x^{2}\) c. Sketch the graph of \(y=\frac{1}{3} x^{2}\) d. Describe the graph of \(y=a x^{2}\) in terms of the graph of \(y=x^{2}\) when \(a > 1\) e. Describe the graph of \(y=a x^{2}\) in terms of the graph of \(y=x^{2}\) when \(0 < a < 1\)

Step-by-Step Solution

Answer

Graph of \( y = ax^2 \) is steeper when \( a > 1 \) and wider when \( 0 < a < 1 \).

Step 1: Sketch the Graph of y = x^2

To sketch the graph of the function \( y = x^2 \), plot points for several values of \( x \). The function is a basic parabolic curve centered at the origin \((0,0)\). Points that can be used include \((0,0), (1,1), (-1,1), (2,4), (-2,4)\). Connect these points to form a U-shaped curve opening upwards.

Step 2: Sketch the Graph of y = 3x^2

For the function \( y = 3x^2 \), the parabola is steeper compared to \( y = x^2 \). To illustrate, calculate a few values like \( (0,0), (1,3), (-1,3), (2,12), (-2,12) \) and plot these points. The graph remains a parabola centered at the origin, but it is narrower and opened upwards because each \( y \)-value is multiplied by 3.

Step 3: Sketch the Graph of y = (1/3)x^2

The graph of \( y = \frac{1}{3}x^2 \) is a parabola that is wider than \( y = x^2 \). Compute values such as \((0,0), (1,1/3), (-1,1/3), (2,4/3), (-2,4/3)\). This time, the parabola opens upwards but each \( y \)-value is one-third of those in \( y = x^2 \), making it less steep.

Step 4: Describe Graph When a > 1

When \( a > 1 \), the graph \( y = ax^2 \) becomes steeper than the graph of \( y = x^2 \). The parabola is narrower because the values of \( y \) increase more rapidly as \( x \) moves away from 0.

Step 5: Describe Graph When 0 < a < 1

When \( 0 < a < 1 \), the graph \( y = ax^2 \) becomes wider compared to \( y = x^2 \). The parabola is less steep as the \( y \) values increase more slowly, producing a wider curve.

Key Concepts

Parabola

A parabola is a symmetrical, U-shaped curve that appears in various mathematical contexts, particularly when dealing with quadratic functions. One of the simplest forms of a parabola is represented by the equation \( y = x^2 \). This specific parabola is centered at the origin, \((0, 0)\), and has its vertex, the highest or lowest point depending on the direction in which it opens, at this point. For \( y = x^2 \), the parabola opens upwards, with the arms extending infinitely as they move away from the vertex.
To sketch a basic parabola like \( y = x^2 \), you can plot several critical points. Start with the origin \((0, 0)\) and then move to points like \((1, 1)\), \((-1, 1)\), and more such as \((2, 4)\) and \((-2, 4)\). These points help illustrate the symmetrical nature of the curve. The further the points from the origin, the higher the value of \( y \) for positive \( x \), and the lower for negative \( x \), indicating the general parabola shape.

Graph Transformations

Graph transformations involve modifying the appearance of a graph based on changes in its function's equation. In the case of a parabola, transformations can alter its width, direction, and position. When considering transformations, changes often occur due to altering coefficients or adding terms to the quadratic function formula.
One way you might transform a parabola is by adjusting the coefficient \( a \) in \( y = ax^2 \):

• If \( a > 1 \), the parabola becomes steeper or narrower. This means for the same \( x \) values, the \( y \) values will be larger, creating a tighter curve around the origin.
• If \( 0 < a < 1 \), the parabola becomes wider. The \( y \) values are smaller for given \( x \) values compared to the normal \( y = x^2 \), meaning the curve spreads out more.
• If \( a < 0 \), the parabola flips to open downwards, though we'll focus mainly on upward-opening cases here.

These transformations allow one to predict and adjust the behavior of the parabola based on real-world data or modeled scenarios.

Vertex Form

The vertex form of a quadratic function offers a convenient way to express the equation of a parabola, making it easier to identify its vertex and axis of symmetry. The vertex form is given by the equation \( y = a(x-h)^2 + k \), where \((h, k)\) represents the vertex of the parabola.
In this form:

• \( a \) influences the parabola's width and direction, similar to what was described under graph transformations. A positive \( a \) keeps the parabola opening upwards, while a negative \( a \) would invert it.
• \( h \) and \( k \) directly indicate the horizontal and vertical shifts of the parabola, respectively. This makes it easier to move the vertex from the origin to any given point \((h, k)\) on the coordinate plane.

Using the vertex form is particularly handy when you want to quickly ascertain the key features of the parabola, like its vertex position, and how steep or wide it is relative to the basic \( y = x^2 \) parabola.