Problem 16 - Algebra and Trigonometry

Question

Find the coordinates of the vertex for the parabola defined by the given quadratic function. $$ f(x)=-2 x^{2}+8 x-1 $$

Step-by-Step Solution

Answer

The coordinates of the vertex for the parabola defined by the given quadratic function are (2, 7).

Step 1: Identify the equations for the vertex from the quadratic function

The general formula for a quadratic function is \(f(x)=a x^{2}+b x+c\). Here, \(a=-2\), \(b=8\) and \(c=-1\). To find the x-coordinate of the vertex\(h\), the formula is \(h=-\frac{b}{2a}\). To find the y-coordinate \(k\), the formula is \(k=f(h)\).

Step 2: Calculate the x-coordinate of the vertex (h)

Plugging the given values into the formula, we get \(h=-\frac{b}{2a} =-\frac{8}{2*(-2)} = 2\). Thus, the x-coordinate of the vertex is 2.

Step 3: Calculate the y-coordinate of the vertex (k)

To find the y-coordinate of the vertex, we substitute the calculated x-coordinate (h=2) into the given quadratic function. So, \(k=f(h) =f(2)=-2*(2)^{2}+8*2-1= -8+16-1 = 7\). Therefore, the y-coordinate of the vertex is 7.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function that has the general form \(f(x) = ax^2 + bx + c\). It is characterized by the squared variable, which gives the graph of the function a distinctive U-shape, known as a parabola. Here, the coefficient \(a\) determines the direction of the parabola (opening upwards if \(a > 0\) and downwards if \(a < 0\)). These functions are fundamental in algebra and are widely used in various real-world applications like calculating projectile paths.

• The term \(bx\) represents the linear component, which influences the position of the vertex along the x-axis.
• The constant \(c\) represents the y-intercept of the parabola, which is where the graph intersects the y-axis.

By understanding the structure of a quadratic function, we can predict the general shape and position of the graph.

Vertex Form

The vertex form of a quadratic function is a different way to express the same parabola, emphasizing its vertex (the peak or lowest point of the curve). It is given by the equation \(f(x) = a(x-h)^2 + k\), where \((h, k)\) are the coordinates of the vertex.

• The vertex form is particularly beneficial because it allows for easy identification of the vertex without completing the square or other algebraic manipulations.
• The parameter \(a\) still indicates the direction and vertical stretch of the parabola, just as it does in the standard quadratic form.

This form is convenient for analyzing and graphing quadratic functions, providing direct information about the vertex and the orientation of the parabola.

Parabola

A parabola is a symmetric curve that is defined as the graph of a quadratic function. One of its key features is that it either opens upward or downward, determined by the coefficient \(a\) in the quadratic function.

• The vertex is the highest or lowest point on the parabola, depending on whether it opens downward or upward, respectively.
• The symmetry of the parabola means that if you fold the graph along the vertical line through the vertex (known as the axis of symmetry), both halves match perfectly.

Parabolas are ubiquitous in multiple fields, including physics, where they represent the paths of projectiles, making their study crucial for understanding natural phenomena.

Coordinates of Vertex

The coordinates of the vertex of a parabola given by a quadratic function \(f(x) = ax^2 + bx + c\) are found using the formula \(h = -\frac{b}{2a}\) for the x-coordinate, and \(k = f(h)\) for the y-coordinate. This provides the critical point \((h, k)\) around which the rest of the parabola is symmetrical.

• The x-coordinate \(h\) determines the line of symmetry and influences the parabola's position on the x-axis.
• The y-coordinate \(k\) tells us how high or low the vertex is on the y-axis, dictating the extremum value (maximum or minimum) the parabola can reach.

Understanding how to determine the vertex is essential for sketching the parabola accurately and for solving problems involving maxima and minima in algebra and calculus.