Problem 15 - College Algebra

Question

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

Step-by-Step Solution

Answer

The vertex of the parabola defined by the quadratic function \(f(x) = -x^2 - 2x + 8\) is at the point \( (1, 5) \).

Step 1: Find Coefficents a, b and c

From the equation, establish that \(a = -1\), \(b = -2\), and \(c = 8\).

Step 2: Compute for the x-coordinate of the vertex

Using the template \(-b/2a\) which is the formula for the x-coordinate of the vertex, substitute 'a' and 'b' from the equation into the formula: \(-( -2)/(2*(-1))\), which simplifies to \(1\).

Step 3: Compute for the y-coordinate of the vertex

Substitute the x-coordinate which is \(1\) into the function \(f(x) = -x^2 - 2x + 8\), it becomes \(-1^2 - 2*1 + 8\), which simplifies to \(5\).

Step 4: Write down the vertex of the parabola

Combine the x-coordinate and y-coordinate to form the vertex of the parabola. So the vertex of the parabola is \( (1, 5) \).

Key Concepts

Quadratic Functions

Quadratic functions are a class of polynomial functions where the highest degree (or power) of the variable is two. They are written in the standard form of f(x) = ax^2 + bx + c, where a, b, and c are coefficients and a is not equal to zero. The graph of a quadratic function is a curve called a parabola.

Every parabola has a vertex, which is the highest or lowest point on the graph, depending on whether the parabola opens upwards (if a > 0) or downwards (if a < 0). The vertex is a significant feature because it represents the maximum or minimum value of the quadratic function. Understanding quadratic functions is crucial when solving various problems in algebra, physics, and engineering where relationships involving squared terms occur.

Vertex Formula

To find the vertex of a parabola represented by a quadratic function, we use the vertex formula. The x-coordinate of the vertex can be found using the equation x = -b/(2a). Here, a and b are the same coefficients found in the standard form of the quadratic equation. Once the x-coordinate is determined, it can be substituted back into the original equation to find the corresponding y-coordinate. Therefore, the y-coordinate of the vertex is f(x) when x is the value found from the vertex formula. This two-step calculation allows for a precise determination of the vertex, providing crucial information about the parabola's shape and location on the coordinate plane.

Graphing Parabolas

Graphing a parabola involves plotting the curve of a quadratic function onto a coordinate plane. The process begins by identifying key characteristics of the parabola, such as the direction in which it opens, its width, and its vertex. The axis of symmetry, a vertical line that passes through the vertex, is also an important aspect to consider while graphing.

To graph a parabola accurately, one must calculate several points on the curve by choosing values for x and solving for y using the quadratic function. The vertex, calculated using the vertex formula, is the most critical point. Other points symmetrical about the axis of symmetry should be plotted to ensure the parabola is shaped correctly. The process of graphing helps visually demonstrate the properties and behavior of quadratic functions.

Coefficients in Quadratic Equations

The coefficients in a quadratic equation, typically labeled a, b, and c, play a pivotal role in defining the characteristics of a parabola. The coefficient a determines the direction and width of the parabola — a wider parabola for a smaller absolute value of a and a narrower one for a larger absolute value. The coefficient b influences the location of the vertex along the x-axis, while c represents the y-intercept, the point where the parabola crosses the y-axis.

Understanding the influence of each coefficient is essential in predicting how the parabola will look before it is graphed. It reveals the key properties of the quadratic function, such as the orientation, shape, and position of the parabola.