Problem 4 - Algebra 1

Question

Find the coordinates of the vertex of the graph of \(y=2 x^{2}+4 x-2\)

Step-by-Step Solution

Answer

The vertex of the graph of the function \(y=2x^{2}+4x-2\) is (-1, -4)

Step 1: Identify a, b and c from the equation

From the equation \(y=2x^{2}+4x-2\), we identify that \(a = 2\), \(b = 4\), and \(c = -2\) respectively.

Step 2: Calculate the x-coordinate of the vertex

The x-coordinate of the vertex h is given by \(h = -b/(2a)\). Substituting the values of a and b from step 1, we find \(h = -4/(2*2) = -1\). So, the x-coordinate of the vertex is -1.

Step 3: Calculate the y-coordinate of the vertex

The y-coordinate of the vertex k is determined by substituting h into the equation. So, \(k = y(h) = 2*(-1)^2 + 4*(-1) - 2 = 2 - 4 - 2 = -4\). Therefore, the y-coordinate of the vertex is -4

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental part of algebra and involve expressions where the highest degree of any term is two. This means the equation will have a general form of:

• \( ax^2 + bx + c = 0 \)

Here, \( a \), \( b \), and \( c \) are coefficients, with \( a eq 0 \). The quadratic formula is typically used to solve these equations, helping to find the roots or solutions at which the quadratic equals zero. The solutions can be real or complex numbers.
Another key feature of quadratic equations is the parabola, a curve that is U-shaped and can open upwards or downwards. The sign of the coefficient \( a \) determines the direction in which the parabola opens:

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

Quadratic equations not only provide solutions but also help in finding other properties such as the axis of symmetry and the vertex, which are essential in the coordinate geometry of parabolas.

Coordinate Geometry

Coordinate geometry, often referred to as analytic geometry, involves representing geometric shapes algebraically using the Cartesian coordinate system. In this system, each point has an \( (x, y) \) coordinate that defines its location in the plane.
For parabolas described by quadratic equations, we can efficiently determine important points like the vertex using coordinates. The coordinate system allows for visualization of the parabola as well as the calculation of distances, slopes, and angles.
Calculating the vertex's coordinates involves using the formula for the x-coordinate \( h = -\frac{b}{2a} \) derived from the equation \( ax^2 + bx + c \). Once \( h \) is calculated, it's substituted back into the equation to find the y-coordinate, \( k \). This effectively gives the vertex in the form \((h, k)\), which pinpoints the 'tip' of the parabola, signifying where it changes direction.

Parabola Properties

Parabolas have unique geometric properties that arise from quadratic equations. They are symmetrical and can be shifted, stretched, or compressed. The vertex of a parabola is its most noticeable feature.
Some key properties include:

• Vertex: The vertex forms the turning point of the parabola, either the lowest or highest point depending on the direction it opens.
• Axis of symmetry: This is a vertical line passing through the vertex and divides the parabola into two mirror-image halves. Its equation is \( x = h \).
• Focus and Directrix: While not always needed for basic vertex calculations, the focus is a point inside the parabola that helps define its shape, and the directrix is a line outside it. Each point on the parabola is equidistant from the focus and the directrix.

Understanding these properties allows us to more thoroughly analyze and sketch quadratic equations' graphs, revealing deeper insights into their behavior and solutions.