Problem 85 - Algebra 2

Question

Write each function in vertex form. $$ y=-4 x^{2}+9 $$

Step-by-Step Solution

Answer

The vertex form of the given equation \( y=-4 x^{2}+9 \) is \( y=-4(x-0)^2 + 9 \), or simplified as \( y=-4x^2+9 \)

Step 1: Identify the standard form equation

The standard form of a parabolic equation is \( y=ax^2+bx+c \). In the given equation, \( y=-4 x^{2}+9 \), the values of a, b and c are -4, 0 and 9 respectively.

Step 2: Write the complete square form

The complete square form of the equation is \( y=a(x-h)^2+k \). In our case, the value of h can be determined by \( h=-\frac{b}{2a} \). As in the given equation, the value of b=0, gives h=0. Substituting a, h, k into the complete square form, we get \( y=-4(x-0)^2 + 9 \)

Step 3: Simplify the equation

On simplifying the equation, we get, \( y=-4x^2 + 9 \)

Key Concepts

Understanding Parabolic Equations

A parabolic equation represents a parabola, which is a U-shaped curve found in various mathematical contexts. This curve is defined by a quadratic function, and the general form of a parabolic equation in two dimensions is given as \( y = ax^2 + bx + c \). Here are some key points:

• Vertex: The highest or lowest point of the parabola, depending on whether it opens upwards or downwards.
• Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two symmetric halves.
• Direction: Determined by the coefficient \( a \) — if \( a > 0 \), the parabola opens upwards, and if \( a < 0 \), it opens downwards.

Understanding these components helps in sketching the graph of the parabola and interpreting its properties.
Moreover, parabolas are common in real-world applications like the paths of projectiles and satellite dishes, owing to their reflective properties.

What is Complete Square Form?

The complete square form of a quadratic equation provides a unique way to express the equation, focusing on the vertex of the parabola. The complete square form is written as \( y = a(x-h)^2 + k \). This form offers two critical pieces of information:

• Vertex Coordinates: The vertex of the parabola is given directly by the point \( (h, k) \). This makes it easier to identify and plot the vertex on a graph.
• Translation of Parabola: This form shows how the parabola is shifted from the origin. \( h \) and \( k \) indicate horizontal and vertical shifts, respectively.

Transforming a standard form equation to complete square form involves completing the square, a technique that organizes the quadratic expression into a perfect square trinomial. This method not only facilitates easy graphing of parabolas but also simplifies solving quadratic equations by revealing the vertex.

Exploring the Standard Form Equation

The standard form of a quadratic equation, \( y = ax^2 + bx + c \), lays the foundation for working with parabolas. This form is crucial as it highlights the following components:

• Coefficient \( a \): Influences the parabola's width and direction. A larger \( |a| \) narrows the parabola, while a smaller \( |a| \) widens it.
• Coefficient \( b \): Alongside \( a \), affects the position of the vertex horizontally.
• Constant term \( c \): Represents the y-intercept, where the parabola crosses the y-axis.

Converting from standard form to vertex (complete square) form involves mathematical manipulation.
This process allows clearer interpretation of the graphing attributes of the parabola.
Understanding these forms is essential for algebraic manipulations and graphical representations in mathematics.