Problem 10 - College Algebra

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 vertex form of the function is \( h(x) = 2(x + 2)^2 - 18 \), and the vertex is \((-2, -18)\).

Step 1: Identify the Quadratic Function Form

The given function is in the form of a quadratic expression, which is generally expressed as \( ax^2 + bx + c \). For the given function \( h(x) = 2x^2 + 8x - 10 \), we identify that \( a = 2 \), \( b = 8 \), and \( c = -10 \).

Step 2: Calculate the Vertex Form

To convert the quadratic function to its standard form (vertex form), we need to complete the square. Start by factoring out the coefficient of \( x^2 \), which is 2, from the first two terms:\[ h(x) = 2(x^2 + 4x) - 10 \]Next, complete the square inside the parenthesis. Take half of the coefficient of \( x \), square it, and add and subtract it inside the parenthesis:\[ h(x) = 2(x^2 + 4x + 4 - 4) - 10 \].This simplifies to:\[ h(x) = 2((x + 2)^2 - 4) - 10 \]

Step 3: Simplify to Vertex Form

Expand and simplify the expression:\[ h(x) = 2(x + 2)^2 - 8 - 10 \].Combine the constants:\[ h(x) = 2(x + 2)^2 - 18 \].Now, the function is in standard (vertex) form, which is \( h(x) = a(x - h)^2 + k \), where \( a = 2 \), \( (h, k) = (-2, -18) \).

Step 4: Identify the Vertex

From the vertex form \( h(x) = 2(x + 2)^2 - 18 \), we can directly identify the vertex of the parabola. The vertex is at the point where \( x = -2 \) because of the \( (x + 2)^2 \) term, and \( y = -18 \) from the constant \( -18 \). Therefore, the vertex is \( (-2, -18) \).

Key Concepts

Vertex Form Conversion

Transforming a quadratic equation from its general form to the vertex form is very useful. It helps us easily find crucial information like the vertex of the parabola. In the standard quadratic form, we write it as:

• General form: \[ ax^2 + bx + c \]
• Vertex form: \[ a(x - h)^2 + k \]

The beauty of the vertex form is that it clearly tells us the vertex, represented as \((h, k)\). To convert from the general form to the vertex form, we use a method called 'completing the square'. The main aim of this is to rewrite part of the quadratic expression as a perfect square trinomial.

Completing the Square

Completing the square is a technique to transform a part of a quadratic expression into a perfect square trinomial. This process helps to convert the quadratic function into its vertex form. Let’s break it down step by step using an example:Suppose we start with: \[ ax^2 + bx + c \]First, focus on the part involving \(x^2\) and \(x\) terms. Factor out \(a\) from these terms if \(a\) is not 1: \[ a(x^2 + \frac{b}{a}x) \]Next, take half of the coefficient of \(x\) (that’s \(\frac{b}{a}\)), square this value, and add and subtract it inside the parentheses:

• Half of \(\frac{b}{a}\) is \(\frac{b}{2a}\)
• Square it to get \(\left(\frac{b}{2a}\right)^2\)

With these values, update the equation:\[ a((x + \frac{b}{2a})^2 - (\frac{b}{2a})^2) + c \]After expanding and simplifying, you will arrive at the vertex form. This method makes the calculation smoother and clearer as further seen with the exercise solution.

Vertex Identification

Once the quadratic is in vertex form \[ a(x - h)^2 + k \], it becomes straightforward to identify the vertex of the parabola. The vertex is the point \((h, k)\) where the parabola either reaches its highest or lowest point. Here's how it works:

• From the expression \[ a(x - h)^2 + k \], \(h\) determines the horizontal shift from the origin, and \(k\) changes the vertical position.
• Thus, the vertex for a quadratic \[ a(x + 2)^2 - 18 \] reveals \( (h, k) = (-2, -18) \).

This gives us the exact position of the vertex in the coordinate plane. Recognizing the vertex instantly aids understanding the graph's direction (downwards if \(a\) is negative, upwards if positive) and its symmetry.