Problem 48 - Algebra 2

Question

Write in the form \(y=a(x-h)^{2}+k\) $$ y=-3 x^{2}-18 x-10 $$

Step-by-Step Solution

Answer

The equation in vertex form is \( y = -3(x+3)^2 + 17 \).

Step 1: Identify the current form

The given equation is presented in the standard form which is \( y = ax^2 + bx + c \). Here, \(a = -3\), \(b = -18\), and \(c = -10\).

Step 2: Complete the square

To rewrite the equation in the form \( y = a(x-h)^2 + k \), we need to complete the square. First, factor out the coefficient of \(x^2\) from the quadratic and linear terms:\[ y = -3(x^2 + 6x) - 10 \] Now, to complete the square inside the parentheses, take half of the coefficient of \(x\), square it, and add and subtract it inside the parentheses:\[ = -3(x^2 + 6x + 9 - 9) - 10 \]Simplify to:\[ = -3((x+3)^2 - 9) - 10 \]

Step 3: Expand and simplify

Distribute the \(-3\) through the terms:\[ = -3(x+3)^2 + 27 - 10 \]Now, simplify the constants:\[ = -3(x+3)^2 + 17 \]

Step 4: Present in vertex form

The equation in the vertex form \( y = a(x-h)^2 + k \) is identified as:\[ y = -3(x+3)^2 + 17 \]Here, \(a = -3\), \(h = -3\), and \(k = 17\).

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree two. This means it has the form:

• \( y = ax^2 + bx + c \)
• Where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \)
• The graph of a quadratic equation is a parabola, which can either open upwards \((a > 0)\) or downwards \((a < 0)\)

Quadratic equations are fundamental in many areas of mathematics because they model various real-world situations, such as projectile motion and area calculations. In our exercise, the quadratic equation is given in the standard form, and we aim to rewrite it into vertex form to easily identify features like the vertex, axis of symmetry, and direction of the parabola.

Completing the Square

Completing the square is a method used to transform a quadratic equation from standard form into vertex form. This technique makes it easier to analyze the properties of the quadratic function and is particularly useful for solving quadratic equations or finding their roots.

Here’s how you complete the square:

• Start with a quadratic expression: \( x^2 + bx \)
• Take half of the coefficient of \( x \) (which is \( b \))
• Square this half and add and subtract it within the expression, i.e., \((\frac{b}{2})^2\)
• This transforms the expression into a perfect square trinomial: \( (x + \frac{b}{2})^2 \)

In the exercise, this method was applied to convert the expression \( x^2 + 6x \) into \((x+3)^2 - 9\). Completing the square is a crucial technique because it reveals the vertex form of a quadratic equation, making the vertex of the parabola more evident.

Standard Form

The standard form of a quadratic equation is characterized by the expression \( y = ax^2 + bx + c \). This is one of the most common representations of quadratic equations because it clearly shows the quadratic (\( ax^2 \)), linear (\( bx \)), and constant (\( c \)) terms.

In the standard form:

• \( a \) determines the direction and width of the parabola
• \( b \) influences the vertex's x-coordinate and the axis of symmetry
• \( c \) is the y-intercept of the parabola

However, the standard form doesn't immediately provide the vertex of the parabola, which is why converting it to the vertex form can be advantageous, like in our exercise. The given standard form \( y = -3x^2 - 18x - 10 \) was transformed into the vertex form \( y = -3(x+3)^2 + 17 \) by completing the square to provide a clearer view of the vertex and to simplify graphing the quadratic function.