Problem 16 - College Algebra

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 given parabolic function are (2, 7).

Step 1: Identify the coefficients a, b, c

From the given function \(f(x) = -2x^{2} +8x - 1\), the values of a, b and c can be identified as a=-2, b=8, and c=-1.

Step 2: Find x-coordinate of the vertex

The x-coordinate of the vertex of a parabola defined by a quadratic function \(f(x) = ax^{2} + bx + c\) is given by the formula \(-b/(2a)\). Substituting a = -2 and b = 8 into the formula gives: \[x = -8 /(2*(-2)) = 2\]. Therefore, the x-coordinate of the vertex is 2.

Step 3: Find y-coordinate of the vertex

Now, substitute the x-coordinate into the function to find the corresponding y-coordinate. This gives: \[f(2) = -2*(2)^{2} +8*2 - 1 = -8 +16 -1 = 7\]. Therefore, the y-coordinate of the vertex is 7.

Key Concepts

Parabola

A parabola is a symmetrical, open plane curve that can appear in two orientations: opening upwards or downwards, depending on the leading coefficient in its equation. When you see a graph that is shaped somewhat like the letter "U" or an overturned "U," that's a typical parabolic curve. These shapes are commonly found in equations that look like this: \( f(x) = ax^2 + bx + c \). You'll know the direction of the parabola by the coefficient \( a \):

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), it opens downwards.

Parabolas are prevalent in physics, engineering, and many real-world structures due to their unique properties. They have an axis of symmetry, meaning one side is a mirror image of the other, centered on a vertical line through the parabola's vertex.

Vertex

The vertex is a significant point on a parabola, representing its highest or lowest point. It is the "tip" or "turning point" where the parabola changes direction. The vertex can be found using a simple formula that depends on the parabola's coefficients: \( a \), \( b \), and \( c \) from the quadratic equation.For the parabola given by \( f(x) = ax^2 + bx + c \), the x-coordinate of the vertex can be calculated using the formula \( x = -\frac{b}{2a} \). This will tell you where the vertex lies along the x-axis. Once you have the x-coordinate, plug it back into the original function to find the y-coordinate, completing the vertex coordinates as \((x, y)\). Understanding the vertex helps in sketching the graph of the parabola and determining its maximum or minimum value.

Coordinates

Coordinates are essentially pairs of numbers that help us pinpoint an exact location on a graph. In the context of parabolas, coordinates are crucial when you're trying to find specific points, such as the vertex of the parabola.Coordinates are usually expressed as \((x, y)\), where:

• \( x \) represents the horizontal position on the graph.
• \( y \) represents the vertical position.

For our quadratic function example, the vertex was found to have the coordinates \((2, 7)\). This means that the vertex is located 2 units across the x-axis and 7 units up the y-axis. Understanding coordinates allows you to navigate and chart out where points fall in a two-dimensional space.

Quadratic Formula

The quadratic formula is a tool used for finding the roots of the quadratic equation, but it can also guide us to understand the vertex better. While not directly used to find the vertex, it is related to the parabola shaped by the quadratic equation and helps find where this curve crosses the x-axis.The quadratic formula is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula gives solutions for \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). Though sometimes the formula and vertex calculation can confuse learners, they are all part of understanding how parabolas behave. The discriminant portion \( b^2 - 4ac \) tells whether you'll have real or complex roots, adding depth to comprehending the parabola's intersections with the x-axis.