Problem 45 - Algebra and Trigonometry

Question

\(45-46=\) A quadratic function is given. (a) Use a graphing device to find the maximum or minimum value of the quadratic function \(f,\) correct to two decimal places. (b) Find the exact maximum or minimum value of \(f,\) and compare with your answer to part (a). $$ f(x)=x^{2}+1.79 x-3.21 $$

Step-by-Step Solution

Answer

The exact minimum of the function is approximately -4.01.

Step 1: Understanding the Form of a Quadratic Function

A quadratic function is generally in the form of \( f(x) = ax^2 + bx + c \). Here, we have \( f(x) = x^2 + 1.79x - 3.21 \). Thus, \( a = 1 \), \( b = 1.79 \), and \( c = -3.21 \). Since \( a > 0 \), the parabola opens upwards, indicating there will be a minimum point.

Step 2: Using Graphing Device for Approximate Minimum

Using a graphing calculator or graphing software, plot the function \( f(x) = x^2 + 1.79x - 3.21 \). Observe the graph to determine the vertex, which is the minimum point since the parabola opens upwards. Estimate the \( x \)-coordinate and \( f(x) \) value of this point to two decimal places.

Step 3: Finding the Exact Minimum Algebraically

For any quadratic function \( ax^2 + bx + c \), the vertex \( x \)-coordinate is given by \( x = -\frac{b}{2a} \). Substituting \( a = 1 \) and \( b = 1.79 \), \[ x = -\frac{1.79}{2 \times 1} = -0.895 \].

Step 4: Computing the Exact Minimum Value of the Function

Substitute \( x = -0.895 \) back into the function to find \( f(x) \). Compute:\[ f(-0.895) = (-0.895)^2 + 1.79(-0.895) - 3.21 \]Calculate each term:\((-0.895)^2 = 0.801025,\)\(1.79(-0.895) = -1.60005,\)\(\therefore f(-0.895) = 0.801025 - 1.60005 - 3.21 \approx -4.01\).

Step 5: Comparing Results

Compare the approximate minimum value from the graphing device with the exact value calculated algebraically. They should be quite close if the graphing calculator was accurate to two decimal places. The exact minimum value was calculated to be approximately \(-4.01\).

Key Concepts

Parabola

A parabola is a U-shaped curve that represents the graph of a quadratic function. For the quadratic function given in the exercise, the formula is in the form of \( f(x) = ax^2 + bx + c \). Here, \( a = 1 \), \( b = 1.79 \), and \( c = -3.21 \). The sign of \( a \) determines the direction of the parabola:

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), it opens downwards, creating a maximum point.

In our case, since \( a = 1 \), the parabola opens upwards, indicating there is a minimum point. This shape is symmetric about the vertical line passing through its vertex. Understanding the parabola's orientation helps in identifying whether we are looking for a minimum or a maximum point in the function.

Vertex

The vertex of a parabola is a critical point and is the position of the minimum or maximum value of a quadratic function's graph. For the function \( f(x) = x^2 + 1.79x - 3.21 \), we find the vertex using a formula. The vertex \( x \)-coordinate is calculated as:\[x = -\frac{b}{2a}\]Plugging in the values from our function, \( a = 1 \) and \( b = 1.79 \), we have:\[x = -\frac{1.79}{2 \times 1} = -0.895\]This means the vertex is at \( x = -0.895 \).
This \( x \)-coordinate is then used to find the \( y \)-value (or \( f(x) \)-value) of the vertex by substituting back into the function. The vertex is crucial since it defines the minimum point for this upward-opening parabola.

Minimum Point

A minimum point on the graph of a quadratic function denotes the lowest point of the parabola. For upward-opening parabolas like in our function, the vertex signifies this minimum point.
After finding the \( x \)-coordinate of our vertex to be \( -0.895 \), we substitute it back into the function to find the function's minimum value. This involves the calculation:\[f(-0.895) = (-0.895)^2 + 1.79(-0.895) - 3.21 \]Breaking it down:

• \((-0.895)^2 = 0.801025\)
• \(1.79(-0.895) = -1.60005\)

Adding these, we calculate:
\[f(-0.895) = 0.801025 - 1.60005 - 3.21 \approx -4.01\]
Thus, the exact minimum point of this quadratic function is approximately at \( x = -0.895 \) with the minimum value \( -4.01 \).

Graphing Calculator

A graphing calculator is a tool that visually represents a quadratic function, allowing us to see the parabolic shape and identify key features like the vertex. To find the minimum point for \( f(x) = x^2 + 1.79x - 3.21 \), you can use a graphing calculator to:

• Plot the function over an appropriate range of \( x \) values.
• Observe the point where the curve reaches its lowest position, determining the minimum point visually.
• Estimate the coordinates of the vertex to two decimal places by zooming in on the graph.

This graphical approximation is then compared with the precise algebraic determination of the vertex, giving you a confirmation of the calculations. Using a graphing calculator provides a quick, intuitive way to understand the behavior of quadratic functions.