Problem 11 - College Algebra with Corequisite Support

Question

For the following exercises, rewrite the quadratic functions in standard form and give the vertex. $$ k(x)=3 x^{2}-6 x-9 $$

Step-by-Step Solution

Answer

The function in vertex form is \( 3(x - 1)^2 - 12 \) with vertex at \((1, -12)\).

Step 1: Identify the quadratic function components

The given quadratic function is \( k(x) = 3x^2 - 6x - 9 \). Here, \( a = 3 \), \( b = -6 \), and \( c = -9 \). This function is in the standard quadratic form \( ax^2 + bx + c \).

Step 2: Complete the square

To rewrite the function in vertex form, we complete the square. Start with \( 3x^2 - 6x \). Factor out the 3: \( 3(x^2 - 2x) \).

Step 3: Find the perfect square trinomial

In \( x^2 - 2x \), take half of the coefficient of \( x \) (which is -2), square it, and add/subtract within the parentheses. Half of -2 is -1. Squaring yields 1, so we have \( 3(x^2 - 2x + 1 - 1) \).

Step 4: Rewrite using perfect square

Rearrange the term to a perfect square minus a number: \( 3((x - 1)^2 - 1) \). Distribute the 3: \( 3(x - 1)^2 - 3 \).

Step 5: Adjust the constant term

Now simplify: \( 3(x - 1)^2 - 9 - 3 = 3(x - 1)^2 - 12 \). The function in vertex form is \( k(x) = 3(x - 1)^2 - 12 \).

Step 6: Identify the vertex

In vertex form, \( k(x) = 3(x - 1)^2 - 12 \), the vertex is at \( (1, -12) \).

Key Concepts

Vertex Form

Quadratic functions can be expressed in different forms, and one of the most helpful is the vertex form.This form reveals the vertex or the highest or lowest point of the parabola described by the quadratic function.
Rewriting the quadratic in vertex form involves expressing it as \( k(x) = a(x - h)^2 + k \), where \((h,k)\) is the vertex of the parabola.

• The coefficient \(a\) indicates the direction of the parabola (up or down) and how "wide" or "narrow" it is.
• The \(h\) and \(k\) values determine the vertex location, which is found directly from the function.

In our exercise, rewriting the function helped find that the vertex is at \((1, -12)\).Knowing where the vertex is can assist in graphing the function quickly and understanding its shape without needing to plot many points.Being able to convert a quadratic from standard form into vertex form is invaluable, especially for graphing purposes and analyzing how the quadratic changes with different parameters.

Completing the Square

Completing the square is a method used to convert the standard quadratic form into a vertex form.This process involves creating a perfect square trinomial so that the expression inside the parentheses forms a square.
Here's how to do it:

• Given a quadratic expression \(ax^2 + bx + c\), start by focusing on the \(ax^2 + bx\) part.
• Factor out the coefficient of \(x^2\) from the first two terms (if needed), making it easier to work inside the parentheses.
• Take half of the coefficient of \(x\), square it, and systematically add and subtract this square inside the expression so you form a complete square trinomial.

Once the square is completed, you can easily convert the expression into the vertex form: \((x-h)^2\), hence uncovering the vertex of the parabola.For our problem, completing the square changed the expression into \(3(x - 1)^2 - 12\), giving us a clearer picture of the parabola's structure.

Standard Form

The standard form of a quadratic function is \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. This is the typical representation of quadratics.This form gives a straightforward look at the coefficients and how they affect the parabola.

• The coefficient \( a \) determines the parabola's direction and width. Positive \( a \) opens upwards, while negative \( a \) opens downwards.
• The \( b \) and \( c \) coefficients influence the location and shape of the parabola, affecting its roots and height.

For the given function, \( k(x) = 3x^2 - 6x - 9 \), it is already presented in standard form with \( a = 3 \), \( b = -6 \), and \( c = -9 \).
Understanding this form is essential as it serves as a starting point for converting into other useful forms, such as vertex form or factored form, to analyze and graph the quadratic function efficiently.