Problem 7 - College Algebra

Question

For the following exercises, rewrite the quadratic functions in standard form and give the vertex. $$ g(x)=x^{2}+2 x-3 $$

Step-by-Step Solution

Answer

The vertex is \((-1, -4)\).

Step 1: Identify the Standard Form

The standard form of a quadratic function is given by: \[ g(x) = a(x - h)^2 + k \]where \((h, k)\) is the vertex of the parabola. We need to rewrite \(g(x) = x^2 + 2x - 3\) into this form.

Step 2: Complete the Square

Start with the expression \(x^2 + 2x\). To complete the square, add and subtract the square of half the coefficient of \(x\) inside the equation:\[ x^2 + 2x = (x^2 + 2x + 1) - 1 = (x + 1)^2 - 1 \]

Step 3: Incorporate the Constant Term

Rewrite the entire function, including the constant term:\[ g(x) = (x + 1)^2 - 1 - 3 \]Simplify the expression:\[ g(x) = (x + 1)^2 - 4 \]

Step 4: Identify the Vertex

The function \( (x + 1)^2 - 4 \) is now in standard form \( g(x) = (x - h)^2 + k \), where \( h = -1 \) and \( k = -4 \). Thus, the vertex of the parabola is \((-1, -4)\).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a key point that gives important information about the shape and position of the parabola on a graph. It is typically denoted as \((h, k)\). In the context of quadratic functions, the vertex is that unique point where the parabola changes direction.

For upward or downward opening parabolas, the vertex is the minimum or maximum point, respectively. The formula for the vertex comes from the standard form equation of a quadratic function, \( g(x) = a(x - h)^2 + k \), where \(h\) and \(k\) dictate the horizontal and vertical shifts from the origin, adjusting the parabola accordingly on the coordinate plane.

Understanding the vertex is crucial because it helps in sketching the graph and understanding the parabola's symmetry. Plus, it assists in determining the range and optimal values of the function depending on whether the parabola opens up or down.

Completing the Square

Completing the square is a mathematical technique used to transform a quadratic expression into a perfect square trinomial, making it easier to analyze and solve. This process is essential for rewriting quadratic functions into their standard form.

Here’s a quick breakdown of how completing the square works using the expression \(x^2+2x\):

• First, take the coefficient of \(x\), which is 2, divide it by 2 to get 1, and then square it to add and subtract it within the expression.
• Transform \(x^2 + 2x\) into \((x^2 + 2x + 1) - 1\), which simplifies to \((x+1)^2 - 1\).

This transformation turns the quadratic expression into a neat square that can be easily structured into the standard form, revealing vital features of the quadratic function.

Completing the square not only aids in identifying the vertex (from the standard form) but also simplifies solving quadratic equations.

Standard Form of a Quadratic Function

The standard form of a quadratic function is one of the most informative ways to express a quadratic equation. It is given by \[ g(x) = a(x - h)^2 + k \],where \(a\) represents the "stretch" of the parabola, and \((h, k)\) is the vertex of the parabola.

This form provides an immediate insight into the parabola's properties:

• \(h\) gives the left or right shift, showing the horizontal position of the vertex.
• \(k\) offers the up or down shift, indicating the vertical position of the vertex.
• \(a\) impacts whether the parabola opens upward (\(a > 0\)) or downward (\(a < 0\)), and how "wide" or "narrow" it appears.

Using the standard form, one can directly read off the vertex and quickly determine the orientation and any shifts of the parabola. This is especially helpful in graphing and analyzing parabolic motion or properties in advanced applications.