Question

What is the \(x\) -coordinate of the vertex of a parabola with the equation \(y=2 x^{2}+3 x-6 ?\) (A) \(\frac{3}{5} \quad \mathrm{OR}\) (B) \(-\frac{1}{4}\)

Short answer

The x-coordinate of the vertex is \(-\frac{3}{4}\).

Step-by-step solution

Identify the Formula for the Vertex

The x-coordinate of the vertex for a parabola in the form \(ax^2 + bx + c\) is given by the formula \(x = \frac{-b}{2a}\).

Determine Values of a and b

In the equation \(y = 2x^2 + 3x - 6\), identify \(a = 2\) and \(b = 3\).

Substitute Values into Vertex Formula

Substitute \(a = 2\) and \(b = 3\) into the vertex formula to calculate the x-coordinate: \(x = \frac{-3}{2 \times 2}\).

Simplify the Expression

Calculate \(x = \frac{-3}{4}\) by simplifying the expression \(\frac{-3}{4}\).

Key concepts

Quadratic Function

A quadratic function is a type of polynomial function where the highest degree, also known as the power or exponent, is two. In simple terms, it is an equation in the form of \( ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). This type of function is characterized by its U-shaped graph, known as a parabola.
Some key points about quadratic functions include:

• The parabola opens upwards if \( a > 0 \) and downwards if \( a < 0 \).
• They always have a vertex, which is the point of maximum or minimum of the function.
• The axis of symmetry is a vertical line that goes through the vertex of the parabola.

The quadratic function plays a crucial role in various real-world applications, including projectile motion and optimization problems.

Vertex Formula

The vertex formula is a critical tool in understanding quadratic functions. The vertex of a parabola, given by its equation in standard form \( ax^2 + bx + c \), can be calculated using the vertex formula. The formula for finding the x-coordinate of the vertex is \( x = \frac{-b}{2a} \).

• This formula provides the x-coordinate of the vertex, which is the point where the parabola either reaches its highest or lowest point.
• By substituting the values of \(a\) and \(b\) from the quadratic function into the vertex formula, you can quickly find this crucial point.

In our exercise, using this formula helps to simplify the process of identifying the vertex, making it a straightforward task to find the coordinates needed to graph or understand the behavior of the quadratic function.

Coordinate Calculation

Calculating the coordinates of the vertex involves a straightforward substitution process using the values of \(a\) and \(b\) from the quadratic equation into the vertex formula. Let's break this down with an example:Given the quadratic equation \( y = 2x^2 + 3x - 6 \), identify:

• \( a = 2 \)
• \( b = 3 \)

Now, substitute these values into the vertex formula \( x = \frac{-b}{2a} \):

• \( x = \frac{-3}{2 \cdot 2} \)
• Simplify to find \( x = \frac{-3}{4} \)

Thus, the x-coordinate of the vertex for this parabola is \(-\frac{3}{4}\). By calculating this coordinate, you can better understand where the vertex lies on the coordinate plane, aiding in graphing or analyzing the quadratic function.