Problem 80 - A Graphical Approach to College Algebra

Question

Write an equation for each parabola with vertex at the origin. Through \((\sqrt{3}, 3) ;\) opening upward

Step-by-Step Solution

Answer

The equation of the parabola is \(y = x^2\).

Step 1: Understand the Parabola Equation Structure

A parabola with its vertex at the origin and opening upward can be described by the equation \( y = ax^2 \). This form assumes the parabola is symmetric around the y-axis.

Step 2: Substitute the Given Point into the Equation

Since the parabola passes through the point \((\sqrt{3}, 3) \), substitute \(\sqrt{3}\) for \(x\) and \(3\) for \(y\) in the equation \( y = ax^2 \) to find the value of \(a\). So, \( 3 = a (\sqrt{3})^2 \).

Step 3: Simplify the Equation

Calculate \((\sqrt{3})^2 = 3\), so the equation becomes \(3 = a \cdot 3\).

Step 4: Solve for \(a\)

Divide both sides of the equation \(3 = 3a \) by 3 to solve for \(a\). Thus, \( a = 1 \).

Step 5: Write the Final Equation

Substitute \(a = 1\) back into the equation \(y = ax^2\), giving us the final equation \(y = x^2\).

Key Concepts

Vertex Form

In the study of parabolas, the vertex form of a quadratic equation is one of the most revealing and intuitive ways to represent a parabola. This form emphasizes the vertex of the parabola, which is either its highest or lowest point. In vertex form, a quadratic equation is expressed as \[ y = a(x-h)^2 + k \] where

• \( a \) represents the vertical stretch or compression of the parabola. If \( a \) is positive, the parabola opens upwards; if negative, it opens downwards.
• \( (h, k) \) is the vertex of the parabola. In scenarios where the vertex is at the origin, \( h = 0 \) and \( k = 0 \), simplifying the equation to \( y = ax^2 \).

Understanding the vertex form is crucial, especially when dealing with transformations of parabolas and comprehending their orientation and position on the coordinate plane.

Quadratic Function

Quadratic functions are fundamental in algebra and appear frequently in various applications. These functions are represented by polynomial equations of degree two. The most common form of a quadratic function is the standard form: \[ y = ax^2 + bx + c \] where

• \( a, b, \) and \( c \) are constants.
• The coefficient \( a \) determines the direction (upward or downward) and the width of the parabola.
• If \( a > 0 \), the parabola opens upwards. If \( a < 0 \), it opens downwards.

Recognizing that quadratics form parabolas, and understanding their properties, helps in graphing these functions and solving equations involving them. Their distinctive U-shape makes them easy to identify and analyze.

Solving for Parameters

Finding specific parameters in a quadratic equation is a common procedure when more information about the parabola is given, such as a point through which it passes. To solve for parameters like \( a \) in our situation where the vertex is at the origin:1. Start with the general formula: \( y = ax^2 \). This simplifies because the vertex is at (0, 0).2. Insert the known point on the parabola, say \((x, y)\).3. For example, using the point \((\sqrt{3}, 3)\), substitute into the equation: \( 3 = a(\sqrt{3})^2 \).4. Solve for \( a \) by simplifying and isolating it.A firm grasp of substituting points and manipulating equations is essential for accurately solving for these parameters, which ultimately define the specific shape and position of the parabola.

Origin in Parabolas

When the vertex of a parabola is at the origin, it simplifies the equation significantly. The vertex at the origin, (0, 0), means that both horizontal and vertical shifts (h and k in the vertex form) are zero.As a result, the equation of the parabola simplifies to \[ y = ax^2 \] This makes calculations and graphing more straightforward because:

• There's no need to adjust for shifts horizontally or vertically.
• The entire focus of the equation is on the coefficient \( a \), which dictates the openness and direction of the parabola.

Using this simple form helps focus on understanding the nature of the parabola itself, making it easier to solve problems involving such basic parabolas.