Problem 24 - Algebra and Trigonometry

Question

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Sketch its graph. (c) Find its maximum or minimum value. $$ f(x)=x^{2}-8 x+8 $$

Step-by-Step Solution

Answer

The function in standard form is \( f(x) = x^2 - 8x + 8 \). Its minimum value is \(-8\). The graph is a parabola opening upwards.

Step 1: Identify the standard form of a quadratic function

The standard form for a quadratic function is given by \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. It is important to know this form to understand and manipulate the function.

Step 2: Express the given function in standard form

In the given function \( f(x) = x^2 - 8x + 8 \), the coefficients are already in the standard form. Therefore, \( a = 1 \), \( b = -8 \), and \( c = 8 \).

Step 3: Sketch the graph of the function

To sketch the graph, note that the parabola opens upwards since \( a = 1 > 0 \). Find the vertex using the vertex formula: \( x = -\frac{b}{2a} = -\frac{-8}{2 \times 1} = 4 \). Substitute \( x = 4 \) into the function to find \( f(4) = (4)^2 - 8(4) + 8 = -8 \). The vertex is \((4, -8)\). The y-intercept is \((0, 8)\). The axis of symmetry is \(x = 4\). Plot the vertex and a few points on either side to sketch the graph.

Step 4: Determine the maximum or minimum value

Since the parabola opens upwards (\( a > 0 \)), the vertex \((4, -8)\) is a minimum point. Thus, the minimum value of the function is \(-8\). The function does not have a maximum value.

Key Concepts

Standard Form of Quadratic Function

Quadratic functions are essential in algebra and are typically expressed in the standard form: \( f(x) = ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are coefficients, with \( a \) being the leading coefficient. What makes quadratic functions unique is the curve of their graph, called a parabola.
Understanding the standard form is crucial because it allows you to quickly identify these coefficients and gives insights into the parabola's shape and direction. For example:

• If \( a > 0 \), the parabola opens upwards, resembling a smile.
• If \( a < 0 \), the parabola opens downwards, more like a frown.

In the function \( f(x) = x^2 - 8x + 8 \), it fits neatly into the standard form with \( a = 1 \), \( b = -8 \), and \( c = 8 \). Recognizing the coefficients is the first step to further analysis, such as graphing or finding the vertex.

Vertex of a Parabola

The vertex of a parabola is its peak or trough point, where it changes direction. Knowing how to find the vertex is key to understanding the graph's shape. For any quadratic function in standard form \( ax^2 + bx + c \), the \( x \)-coordinate of the vertex can be calculated using the formula:

• \( x = -\frac{b}{2a} \)

For our function \( f(x) = x^2 - 8x + 8 \), we find:

• \( x = -\frac{-8}{2 \cdot 1} = 4 \)

Substituting \( x = 4 \) back into the function to find the \( y \)-coordinate, we get:

• \( f(4) = 4^2 - 8 \times 4 + 8 = -8 \)

Thus, the vertex is \((4, -8)\). This point is valuable as it divides the parabola into two symmetric halves and is key when sketching the function on a graph.

Maximum or Minimum Value of a Quadratic Function

Quadratic functions exhibit either a maximum or minimum value, depending on the direction of the parabola. This value appears at the vertex of the parabola.Since our given function \( f(x) = x^2 - 8x + 8 \) has \( a = 1 > 0 \), the parabola opens upwards. This upward opening indicates that the vertex represents the minimum point on the graph. Therefore, there is no maximum value due to the parabola's shape extending infinitely upwards.
The minimum value, located at its vertex, is the function's lowest point. For \( f(x) = x^2 - 8x + 8 \), the minimum value occurs at the vertex \((4, -8)\), making the minimum value \(-8\). In real-world terms, this could represent a lowest cost, shortest time, or other minimum measures, depending on the problem context.