Problem 22 - Algebra and Trigonometry

Question

19-28 \(=\) 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 minimum value of the function is -8, occurring at \(x = 4\).

Step 1: Identify the Coefficients

First, let's identify the coefficients from the quadratic function \(f(x) = x^2 - 8x + 8\). The coefficients are: \(a = 1\), \(b = -8\), and \(c = 8\).

Step 2: Express the Quadratic in Standard Form

The quadratic function \(f(x) = x^2 - 8x + 8\) is given in standard form. The standard form of a quadratic function is \(f(x) = ax^2 + bx + c\). Here, the given function already matches this form with \(a = 1\), \(b = -8\), and \(c = 8\).

Step 3: Find the Vertex Form

To find the vertex and determine if the graph has a maximum or minimum, use the vertex formula \(x = -\frac{b}{2a}\). For the function \(f(x) = x^2 - 8x + 8\), calculate \(x = -\frac{-8}{2 \times 1} = 4\). Substitute \(x = 4\) back into the function to find \(f(4)\): \(f(4) = (4)^2 - 8 \times 4 + 8 = -8\). Thus, the vertex is \((4, -8)\).

Step 4: Determine the Nature of the Vertex

Given the coefficient \(a = 1\) is positive, the parabola opens upwards, meaning the vertex \((4, -8)\) is a minimum point. Thus, the minimum value of the function is \(-8\).

Step 5: Sketch the Graph

To sketch the graph, plot the vertex \((4, -8)\). Since the parabola opens upwards, draw a U-shaped curve centering on the vertex. Use additional points if necessary to guide the shape, such as intercepts: \(f(0) = 8\), giving the y-intercept at (0, 8), and solve for the x-intercepts if needed.

Key Concepts

Standard Form of a Quadratic Function

A quadratic function is expressed in the form \[f(x) = ax^2 + bx + c\], where

• \(a\), \(b\), and \(c\) are coefficients
• \(a\) cannot be zero,

The equation \(f(x) = x^2 - 8x + 8\) fits into this format seamlessly, thus representing the standard form of a quadratic function.
Being in this form makes it easier to recognize the various components of the function, which are crucial for understanding its behavior and shape:

• The coefficient \(a = 1\) determines how "wide" or "narrow" the parabola is.
• The coefficient \(b = -8\) influences the symmetry of the parabola.
• The constant \(c = 8\) gives the y-intercept of the graph.

Recognizing these elements helps predict the general shape of the parabola and supports further transformation to find other forms or characteristics of the quadratic function.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by \[f(x)= a(x-h)^2 + k\], where

• \((h, k)\) is the vertex of the parabola,
• \(h\) represents the x-coordinate of the vertex,
• \(k\) represents the y-coordinate, giving the maximum or minimum value.

For the function \(f(x) = x^2 - 8x + 8\), you can determine the vertex using the formula \(x = -\frac{b}{2a}\), where the solution for \(x\) gives us the x-coordinate of the vertex.
By substituting the identified values of \(a = 1\) and \(b = -8\), we calculate:\[x = -\frac{-8}{2 \times 1} = 4\]Plugging \(x = 4\) back into the original function develops the complete vertex coordinate:\[k = f(4) = (4)^2 - 8\times4 + 8 = -8\]This reveals that the vertex is at \((4, -8)\), implying that the function can be rewritten as \[f(x) = (x - 4)^2 - 8\].
This form is particularly useful as it clearly highlights the turning point of the quadratic function.

Maximum and Minimum Values in Quadratics

Quadratic functions reach a maximum or minimum value at their vertex. The coefficient \(a\) in the standard form determines the direction of the parabola:

• If \(a > 0\), the parabola opens upwards, indicating that the vertex is a minimum point.
• If \(a < 0\), the parabola opens downwards, so the vertex represents a maximum.

For our function \(f(x) = x^2 - 8x + 8\), with \(a = 1\) being positive, the parabola opens upwards, and consequently, the vertex \((4, -8)\) signifies the minimum point.
This insight helps not only in pinpointing the vertex but also in understanding the behavior of the entire graph.The minimum value of the function is \(-8\), which occurs when \(x = 4\).
This property is particularly helpful when tackling optimization problems, as it allows you to readily identify extreme values courtesy of the vertex.