Problem 10 - College Algebra with Corequisite Support

Question

For the following exercises, rewrite the quadratic functions in standard form and give the vertex. $$ h(x)=2 x^{2}+8 x-10 $$

Step-by-Step Solution

Answer

The standard form is \(h(x) = 2(x+2)^2 - 18\), with vertex \((-2, -18)\).

Step 1: Identify the Quadratic Function

The given quadratic function is \( h(x) = 2x^2 + 8x - 10 \). The goal is to rewrite this function in standard form, which is \( h(x) = a(x-h)^2 + k \).

Step 2: Complete the Square

To convert the quadratic equation into standard form, we complete the square. First, factor out the leading coefficient (2) from the quadratic and linear terms:\[ h(x) = 2(x^2 + 4x) - 10\]Next, take half of the coefficient of \(x\), square it, and add and subtract this value inside the parenthesis. Half of 4 is 2, and 2 squared is 4:\[ h(x) = 2(x^2 + 4x + 4 - 4) - 10\]Simplify inside the parenthesis:\[ h(x) = 2((x+2)^2 - 4) - 10\]

Step 3: Simplify the Expression

Distribute the 2 to both terms in the parenthesis:\[ h(x) = 2(x+2)^2 - 8 - 10\]Simplify the constant terms:\[ h(x) = 2(x+2)^2 - 18\] This is the standard form of the quadratic function.

Step 4: Identify the Vertex

The standard form of the quadratic function is \( h(x) = a(x-h)^2 + k = 2(x+2)^2 - 18 \). The vertex \((h, k)\) is given by \((-2, -18)\).

Key Concepts

Standard Form

A quadratic function is often given in the format of ax^2 + bx + c. However, it can be more insightful to express it in what is known as "standard form." Standard form is represented by \( h(x) = a(x-h)^2 + k \). In this version, you can instantly identify the vertex, which we will discuss next. Knowing how to convert a function into standard form is valuable because it shows whether the parabola opens upwards or downwards:

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

Understanding this concept helps in sketching graphs and solving quadratic equations.

Vertex

The vertex of a quadratic function is a crucial point, representing the peak or the lowest point of a parabola. It provides important information and is easy to find once the function is in standard form. The vertex \(h, k\) in \(a(x-h)^2 + k \) can be read directly from the function:

• "h" is the x-coordinate of the vertex.
• ”k” is the y-coordinate.

For example, if you have the quadratic function \( h(x) = 2(x+2)^2 - 18 \), the vertex is found by locating \((h, k)\) where \(h = -2\) and \(k = -18\). This means the parabola reaches its lowest point at (-2, -18) because \( a > 0 \), indicating the parabola opens upwards.

Completing the Square

Completing the square is a method used to transform a quadratic equation into its standard form. This technique involves several steps, which make deeper understanding and practice key.Let's break down the process using an example from our original exercise.

• First, factor out the leading coefficient from the quadratic and linear terms if it is not equal to 1, as seen: \( h(x) = 2(x^2 + 4x) - 10 \).
• Take half the coefficient of the x-term, square it, and both add and subtract it inside the parentheses to maintain the equality: \( 2(x^2 + 4x + 4 - 4) - 10 \).
• Simplify by grouping into perfect square form: \( 2((x+2)^2 - 4) \).
• Expand and simplify: \( 2(x+2)^2 - 8 - 10 \) simplifies to \( 2(x+2)^2 - 18 \).

By completing the square, the function can be rewritten in standard form, making it easy to extract important properties such as the vertex.