Problem 42 - Introductory Algebra for College Students

Question

Find the vertex for the parabola whose equation is given by first writing the equation in the form \(y=a x^{2}+b x+c\) $$y=2(x-1)^{2}-4$$

Step-by-Step Solution

Answer

The vertex of the given parabola is at point (1, -4).

Step 1: Identify the Vertex Form of the Parabola

The vertex form of the parabola is \(y=a(x-h)^{2}+k\). So the vertex of the parabola is at point (h, k).

Step 2: Compare with the Given Equation

Now, compare the given equation \(y=2(x-1)^{2}-4\) with the vertex form. It can be observed that the equation already is in vertex form.

Step 3: Identify the Vertex

From step 2, identify the values of h and k. Here, h=1 and k=-4. Therefore, the vertex of the parabola is at point (1, -4).

Key Concepts

Parabola Equations

A parabola is a U-shaped graph that is represented in a coordinate plane by a quadratic equation. The most common form of this equation is the standard form, written as \[y = ax^2 + bx + c.\]Here,

• \(a\), \(b\), and \(c\) are constants.
• \(x\) and \(y\) are variables.

A parabola can open upwards or downwards. The direction is determined by the value of \(a\). If \(a > 0\), the parabola opens upwards, creating a minimum point. If \(a < 0\), the parabola opens downwards, indicating a maximum point.
The vertex of the parabola, which is a key point, is where the direction changes. This point provides valuable information about the curve's minimum or maximum. Understanding the parabola's equation helps in sketching the graph and analyzing the vertex. Each part of the equation plays a crucial role in these calculations.

Vertex Form

The vertex form of a quadratic equation is especially useful for finding the vertex of a parabola quickly and accurately. The equation is written as \[y = a(x-h)^2 + k.\]In this form, the vertex of the parabola is clearly indicated by the point

• \((h, k)\).

Here,

• \(h\) represents the horizontal shift from the origin.
• \(k\) represents the vertical shift.

For example, consider the equation \(y = 2(x-1)^2 - 4\).
By comparing this with the vertex form \(y = a(x-h)^2 + k\), we can easily identify the vertex. In this equation, \(h = 1\) and \(k = -4\), making the vertex (1, -4).
This format simplifies finding the vertex compared to other forms like the standard form.

Quadratic Functions

Quadratic functions are equations of the form \[y = ax^2 + bx + c,\]which graph as parabolas. These functions are fundamental in algebra and appear frequently in various mathematical and real-world applications.
They always have a degree of 2, which differentiates them from linear functions, which are degree 1.
Quadratic functions have several key characteristics:

• They produce a parabolic graph, either opening upwards or downwards.
• The coefficient \(a\) affects the width and direction of the parabola.
• The vertex provides information about either the minimum or maximum value.

Additionally, the graph's symmetry is another important feature, as every parabola is symmetric about its axis of symmetry. This axis is a vertical line running through the vertex, dividing the parabola into two mirror-image halves.Understanding quadratic functions is essential for many mathematical problem-solving scenarios, from calculating projectile motion to optimizing areas or profits.