Question

What is the \(x\) -coordinate of the minimum of the parabola with the equation \(y+17=6 x^{2}+12 x ?\) (A) \(-1\) (B) 0 (C) 2 (D) 3

Short answer

The x-coordinate of the minimum of the parabola is \(-1\), which corresponds to option (A).

Step-by-step solution

Define the standard form of a quadratic equation

The given equation is \( y + 17 = 6x^2 + 12x \). First, rearrange it into the standard form \( y = ax^2 + bx + c \) by subtracting 17 from both sides: \( y = 6x^2 + 12x - 17 \). This shows \( a = 6 \), \( b = 12 \), and \( c = -17 \).

Write the formula to find the x-coordinate of the parabola's vertex

The x-coordinate of the vertex of a parabola \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). This formula gives the x-coordinate of the minimum point for a parabola that opens upwards (when \( a > 0 \)).

Substitute the values of a and b

Substitute the values of \( a = 6 \) and \( b = 12 \) into the vertex formula: \( x = -\frac{12}{2 \times 6} \).

Calculate the x-coordinate of the vertex

Simplify the expression: \( x = -\frac{12}{12} \), which results in \( x = -1 \). This is the x-coordinate of the minimum point of the parabola.

Key concepts

Vertex of a Parabola

In the world of quadratic equations, the vertex of a parabola plays a central role. The vertex is a special point that represents either the highest or lowest point on the graph of a quadratic function. Think of the vertex as the turning point of the parabola. It is where the parabola changes direction if it is opening up or down.

For a parabola described by the quadratic equation in standard form, i.e., \(y = ax^2 + bx + c\), the vertex can be found and depicted using the formula for the x-coordinate of the vertex, which is \(x = -\frac{b}{2a}\). Understanding the vertex helps us analyze the entire graph of the quadratic equation to make predictions about its behavior.

x-coordinate

The x-coordinate of a point tells us the horizontal position of that point on the graph. For the vertex of a parabola, the x-coordinate is crucial because it tells us where the minimum or maximum point occurs along the horizontal axis.

The x-coordinate of the vertex can be calculated by using the formula \(x = -\frac{b}{2a}\), where \(a\) and \(b\) are coefficients from the quadratic equation in standard form \(y = ax^2 + bx + c\). Calculating this value provides insight into the symmetry and position of the parabola's vertex on the graph.

Standard Form

In quadratic equations, the standard form is a canonical way of writing the expression as \(y = ax^2 + bx + c\). This form is particularly helpful because it reveals key parameters of the equation:

• \(a\): The coefficient of \(x^2\), which determines how "open" or "closed" the graph appears
• \(b\): The coefficient of \(x\), which, along with \(a\), influences the position of the vertex
• \(c\): The constant term, which often affects the y-intercept

Working with equations in standard form allows us to apply formulas, like finding the vertex, more easily and accurately. Recognizing the standard form is the first step to solving many problems involving quadratic equations.

Minimum Point

The minimum point of a parabola is the lowest point on its graph, and it occurs when the parabola opens upwards (when the coefficient \(a > 0\)). This point is located at the vertex of the parabola.

In mathematical terms, the minimum point is important because it represents the minimum value that the quadratic function can take. By determining the x-coordinate of the vertex using \(x = -\frac{b}{2a}\), we find where this minimum occurs along the x-axis.

Understanding the concept of the minimum point helps us in various real-life applications, such as finding optimal solutions in optimization problems or understanding natural phenomena.