Problem 83 - Algebra 2

Question

Rewrite each equation in vertex form. $$ y=3 x^{2}-7 $$

Step-by-Step Solution

Answer

The equation in vertex form is: \(y = 3x^{2} - 7\)

Step 1: Analyze the given standard form equation

The given equation is in standard form (quadratic form), i.e., \(y=ax^{2}+bx+c\). In the given equation, \(y = 3x^{2} - 7\), the coefficients are a = 3, b = 0, and c = -7.

Step 2: Identifying the vertex form

The vertex form of a parabola's equation is \(y = a(x - h)^{2} + k\). Where (h,k) are the coordinates of the vertex. Here the task is to transform the given standard form into this form.

Step 3: Determine the vertex (h, k)

Given that the equation is already in the form \(y = a(x - h)^{2} + k\), the vertex (h, k) of the parabola is (0, -7) since b = 0.

Step 4: Rewrite the equation in vertex form

Plug the values of h, k and a back into the vertex form equation to obtain the equation in vertex form: \(y = 3(x - 0)^{2} - 7\)

Step 5: Simplify the equation

Simplify the equation by removing 0 in (x - 0), which results in: \(y = 3x^{2} - 7\)

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions where the highest power of the variable is squared, such as \( ax^2 + bx + c = 0 \). These equations can describe a wide range of real-world phenomena, from the trajectory of a ball to the shape of parabolic dishes in satellite antennas. Let’s break it down:

• "\( a \)" represents the quadratic coefficient and dictates the direction and width of the parabola.
• "\( b \)" is the linear coefficient and affects the position of the vertex along the x-axis.
• "\( c \)" is the constant term that determines the y-intercept, the point at which the graph intersects the y-axis.

Grasping these components helps in manipulating the equation into different forms, such as vertex form, which offers a clearer view of the parabola's geometric features.

Standard Form

The standard form of a quadratic equation is expressed as \( y = ax^2 + bx + c \). This form is advantageous because it provides a straightforward way to compute the roots of the equation using methods like factoring, completing the square, or the quadratic formula.
In the problem provided, the equation \( y = 3x^2 - 7 \) is already in standard form. Here:

• \( a = 3 \)
• \( b = 0 \)
• \( c = -7 \)

Understanding this structure lays the groundwork for further transformations, such as converting to vertex form, which often simplifies graphing tasks.

Parabola

A parabola is a U-shaped curve that can open upwards or downwards, as described by quadratic equations. The shape of the parabola is determined by the sign and value of \( a \):

• If \( a > 0 \), it opens upwards.
• If \( a < 0 \), it opens downwards.
• The absolute value of \( a \) dictates the width, with larger values resulting in a narrower parabola.

In the given problem, the parabola opens upwards since \( a = 3 \). The vertex, which is the peak or trough of the parabola, provides insight into the highest or lowest point on the graph, crucial for various practical applications like maximizing profit or minimizing cost.

Vertex (h, k)

The vertex of a parabola is represented by \((h, k)\), where \(h\) is the x-coordinate and \(k\) is the y-coordinate of the vertex. This point is significant as it represents the turning point of the parabola. In vertex form, the quadratic equation is expressed as \( y = a(x - h)^2 + k \).
To find the vertex from the standard form, especially when \( b = 0 \) as in our example, it becomes much more straightforward:

• \( h \) is directly 0, because the equation can be directly rewritten without the \( bx \) term.
• \( k \) is equal to the constant term \( c \), in this case, \(-7\).

Thus, the vertex for the equation \( y = 3x^2 - 7 \) is at \((0, -7)\), which also happens to be the minimum point of the parabola due to the positive value of \( a \). This vertex form makes it easier to visualize and graph the parabola efficiently.