Problem 55 - Algebra A Combined Function

Question

Find the maximum or minimum value of each function. Approximate to two decimal places. $$ f(x)=2.3 x^{2}-6.1 x+3.2 $$

Step-by-Step Solution

Answer

The minimum value is approximately -0.84.

Step 1: Identify the Quadratic Function Coefficients

The function given is a quadratic in the form of \( ax^2 + bx + c \). Here, \( a = 2.3 \), \( b = -6.1 \), and \( c = 3.2 \).

Step 2: Determine the Nature of the Parabola

Since the coefficient \( a = 2.3 \) is positive, the parabola opens upwards. This means the function has a minimum value at its vertex.

Step 3: Find the Vertex Formula for the Minimum

The x-coordinate of the vertex, which gives the minimum value, can be found using the vertex formula \( x = \frac{-b}{2a} \). Substitute the coefficients to find the x-coordinate of the vertex.

Step 4: Calculate the X-Coordinate of the Vertex

Substitute \( a = 2.3 \) and \( b = -6.1 \) into the formula: \[ x = \frac{-(-6.1)}{2 \times 2.3} = \frac{6.1}{4.6} \approx 1.33 \].

Step 5: Calculate the Minimum Value of the Function

Substitute \( x = 1.33 \) back into the function to find \( f(x) \): \[ f(1.33) = 2.3(1.33)^2 - 6.1(1.33) + 3.2 \].

Step 6: Evaluate the Function at the Vertex

Calculate \( f(1.33) \): \[ f(1.33) = 2.3(1.7689) - 8.113 + 3.2 \approx 4.06847 - 8.113 + 3.2 \approx -0.84 \].

Step 7: Conclude with the Minimum Value

The minimum value of the function \( f(x) = 2.3x^2 - 6.1x + 3.2 \) is approximately \( -0.84 \).

Key Concepts

Vertex Form

Quadratic functions can be rewritten in what is called the "vertex form." This form is particularly helpful because it directly reveals the vertex of the parabola, which is the highest or lowest point of the graph. The vertex form of a quadratic function is given as: \[ f(x) = a(x-h)^2 + k \] where \( (h, k) \) is the vertex of the parabola.

• \( h \) determines the horizontal shift from the origin.
• \( k \) represents the vertical shift.
• \( a \) reflects the width and direction of the parabola.

To convert the standard form to vertex form, one may need to complete the square. For our function, finding the vertex directly using the formula \( x = \frac{-b}{2a} \) is another method to locate the vertex efficiently.

Parabola

A parabola is a U-shaped curve on a graph that is a key feature of quadratic functions. It can open upwards or downwards depending on the coefficient of the \( x^2 \) term. In the given function \( f(x)=2.3x^2-6.1x+3.2 \), the parabola opens upward because:

• The coefficient \( a = 2.3 \) is positive.

This upward opening indicates that the function will have a minimum point. Parabolas have unique properties, like symmetry, with the axis of symmetry passing through their vertex.

• The vertex is either the lowest or highest point depending on the parabola's orientation.

Understanding the direction and vertex of a parabola assists in predicting the behavior of the quadratic function.

Minimum Value

The minimum value of a quadratic function occurs at the vertex if the parabola opens upwards. For our quadratic function, \( f(x) = 2.3x^2 - 6.1x + 3.2 \), a positive \( a \) means the graph opens upwards, and thus, the minimum value exists at its vertex. The formula to find the x-coordinate of the vertex is: \[ x = \frac{-b}{2a} \] Substituting the values gives us \( x \approx 1.33 \).

• Once \( x \) is found, substitute back into the function to find the function's minimum value: \( f(1.33) \approx -0.84 \).
• This minimum value \( -0.84 \) is the lowest point on the graph when \( x = 1.33 \).

The concept of minimum value is critical in various applications, such as optimizing a real-world system or process.

Coefficient

Coefficients in a quadratic function \( ax^2 + bx + c \) significantly influence the shape and position of the parabola. Let's break down the influence of each:

• Coefficient \( a \): Determines the direction (upward or downward) and the width of the parabola. Larger absolute values of \( a \) indicate a narrower parabola.
• Coefficient \( b \): Affects the location of the vertex along the x-axis. The vertex formula \( x = \frac{-b}{2a} \) highlights its role.
• Coefficient \( c \): Provides the y-intercept, which is the point where the parabola crosses the y-axis.

Understanding these coefficients enables us to predict graph behavior and apply transformations efficiently. In our example, \( a = 2.3 \), \( b = -6.1 \), and \( c = 3.2 \) help determine the upward opening and shape of the parabola.