Problem 36 - College Algebra

Question

Determine, without graphing, whether the given quadratic function has a minimum value or \(a\) maximum value. Then find the coordinates of the minimum or the maximum point. \(f(x)=2 x^{2}-8 x-3\)

Step-by-Step Solution

Answer

The function \(f(x)=2x^{2}-8x-3\) has a minimum value. The coordinates of the minimum point are \((2,-7)\).

Step 1: Identify the type of extremum

Check the coefficient of \(x^2\) in the function \(f(x)=2x^{2}-8x-3\). As the coefficient \(2\) is positive, the function has a minimum value.

Step 2: Find x-coordinate of the extremum point

Use the formula \(-b/(2a)\) to find the x-coordinate of the minimum point. Here, \(a=2\) and \(b=-8\), hence the x-coordinate of the minimum point is \(-(-8)/(2*2)=2\).

Step 3: Find y-coordinate of the extremum point

Insert this value into the function to find the corresponding y-value. \(f(2)=2*2^{2}-8*2-3=-7\). Thus, the y-coordinate of the minimum point is -7.

Key Concepts

Extremum Points

Extremum points in a quadratic function are where the function reaches its highest or lowest value. These points are critical for understanding the shape and behavior of the parabola. Quadratic functions generally take the form \( f(x) = ax^2 + bx + c \). The coefficient \( a \) determines the parabola's orientation:

• If \( a > 0 \), the parabola opens upwards, indicating the function has a minimum point.
• If \( a < 0 \), the parabola opens downwards, indicating the function has a maximum point.

In the given quadratic function \( f(x) = 2x^2 - 8x - 3 \), the coefficient \( a \) is 2, which is positive. Therefore, the parabola opens upward, suggesting that the function has a minimum point. These extremum points are also referred to as the vertex of the parabola.

Vertex of a Parabola

The vertex of a parabola in a quadratic function is its point of extremum. For the function \( f(x) = ax^2 + bx + c \), the vertex can be found using the formula for the x-coordinate: \( x = \frac{-b}{2a} \). This formula gives us a straightforward way to find where the minimum or maximum occurs.For the function \( f(x) = 2x^2 - 8x - 3 \), we can identify \( a = 2 \) and \( b = -8 \). Plugging these into the formula yields:\[x = \frac{-(-8)}{2 \times 2} = 2.\]Once we have the x-coordinate, we can substitute it back into the function to find the y-coordinate:\[f(2) = 2(2)^2 - 8(2) - 3 = -7.\]Thus, the vertex of the parabola, which is the minimum point, is at the coordinates \( (2, -7) \). This point is crucial in describing the position and direction of the parabola on a coordinate plane.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is crucial for visualizing and solving problems involving parabolas and other geometric shapes on a coordinate plane. In the context of quadratic functions, coordinate geometry helps in understanding the position of extremum points and the shape of the parabola.For the quadratic equation like \( f(x) = 2x^2 - 8x - 3 \), coordinate geometry provides tools to:

• Plot the parabola based on the quadratic coefficients \( a \), \( b \), and \( c \).
• Locate the extremum (vertex), which offers insights into whether the parabola opens upwards or downwards, and where its highest or lowest point is.

By understanding how to use coordinate geometry, one can easily interpret the features of any quadratic function. It becomes possible to accurately pinpoint vertices, intercepts, and other key attributes of parabolas on a two-dimensional plane, which is invaluable for deeper algebraic and geometric studies. This comprehensive understanding can make solving quadratic function problems much more intuitive and effective.