Problem 50 - Algebra and Trigonometry

Question

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 it with your answer to part (a). $$ f(x)=1+x-\sqrt{2} x^{2} $$

Step-by-Step Solution

Answer

Maximum value of the function is \(1 + \frac{\sqrt{2}}{8}\), approximately 1.18.

Step 1: Determine the Type of Quadratic

Examine the quadratic function \( f(x) = 1 + x - \sqrt{2} x^2 \). Since the coefficient of \( x^2 \) is negative \((-\sqrt{2})\), this function opens downwards, indicating that it has a maximum value.

Step 2: Calculate the Vertex Using the Formula

The vertex \((h, k)\) of a quadratic function \( f(x) = ax^2 + bx + c \) is found using the formula \( x = -\frac{b}{2a} \).Substitute \( a = -\sqrt{2} \) and \( b = 1 \):\[ x = -\frac{1}{2(-\sqrt{2})} = \frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4} \] by rationalizing the denominator.

Step 3: Calculate Maximum Value

Substitute \( x = \frac{\sqrt{2}}{4} \) back into the function to find the maximum value:\[f\left(\frac{\sqrt{2}}{4}\right) = 1 + \frac{\sqrt{2}}{4} - \sqrt{2}\left(\frac{\sqrt{2}}{4}\right)^2\]\[f\left(\frac{\sqrt{2}}{4}\right) = 1 + \frac{\sqrt{2}}{4} - \sqrt{2}\left(\frac{1}{8}\right)\]\[f\left(\frac{\sqrt{2}}{4}\right) = 1 + \frac{\sqrt{2}}{4} - \frac{\sqrt{2}}{8}\]\[f\left(\frac{\sqrt{2}}{4}\right) = 1 + \frac{\sqrt{2}}{8}\]Thus, the exact maximum value is \(1 + \frac{\sqrt{2}}{8}\).

Step 4: Graph the Function (Approximation)

Using a graphing device, plot \( f(x) = 1 + x - \sqrt{2} x^2 \). Observing the graph, it should achieve a maximum value near the vertex. For precision, use the graphing device to find a maximum to two decimal places. It should show a value close to 1.18.

Step 5: Compare Exact and Approximate Values

Compare the exact maximum value, \(1 + \frac{\sqrt{2}}{8}\), with the approximated value from the graphing device. \(1 + \frac{\sqrt{2}}{8}\) is approximately equal to 1.18, confirming the precision of our calculation.

Key Concepts

Maximum and Minimum Values

In a quadratic function, maximum and minimum values play a key role. A quadratic function typically has the form \( f(x) = ax^2 + bx + c \). Depending on the sign of the coefficient \( a \), a quadratic function can either open upwards or downwards. If \( a \) is positive, the parabola opens upwards and has a minimum value at its vertex. Conversely, if \( a \) is negative, as in the case of \( f(x) = 1 + x - \sqrt{2} x^2 \), the function opens downwards, indicating a maximum value at the vertex.

• Maximum values occur when the parabola is downward-opening.
• Minimum values occur when the parabola is upward-opening.

These values are crucial for understanding the behavior of quadratic functions, especially in applications like physics, economics, and engineering, where determining the limits of a function can inform decisions and predictions.

Vertex of a Parabola

The vertex of a parabola is a pivotal point that gives us the peak or trough of the quadratic curve. It is essential for understanding where the maximum or minimum value occurs in a quadratic function. The vertex is given by the formula for a function \( f(x) = ax^2 + bx + c \) as:\[ x = -\frac{b}{2a} \].
For our specific function, \( f(x) = 1 + x - \sqrt{2} x^2 \), we calculate\[ x = -\frac{1}{2(-\sqrt{2})} = \frac{\sqrt{2}}{4} \].
The x-coordinate of the vertex allows us to find the corresponding value of the function, which will be the highest or lowest point on a graph, depending on the function's orientation.

• The x-coordinate is found using the formula \( x = -\frac{b}{2a} \).
• The y-coordinate is calculated by substituting this x-value back into the original function.

Finding the vertex not only helps in determining the maximum or minimum but also in sketching the graph accurately. By understanding its position, one can see how the quadratic curve aligns on a graph.

Graphing Quadratics

Graphing quadratics is a skill that brings the algebraic function to life visually, helping to better understand its key features. The quadratic formula defines a parabola and can be visualized by plotting the function on an x and y-axis.
When graphing, note the direction: a positive \( a \) value means the parabola faces upwards, whereas a negative one faces downwards, as in our function \( f(x) = 1 + x - \sqrt{2} x^2 \).
Here are some steps to graph a quadratic function:

• Identify the direction of the parabola by checking the sign of \( a \).
• Calculate the vertex and plot it on your graph.
• Determine additional points by choosing x-values and calculating corresponding y-values.
• Draw a smooth curve through all points.

Using technology, such as a graphing calculator, can aid in finding the exact vertex and maximum or minimum value effectively. A visual representation not only helps in problem-solving but also solidifies the understanding of the function's properties.

Quadratic Formula

The quadratic formula is a reliable method for finding the roots of any quadratic equation of the form \( ax^2 + bx + c = 0 \). This formula is stated as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
While this formula is not directly involved in finding maximum or minimum values, it provides insight into the x-intercepts of the function, which are crucial in graph analysis.
Steps involved in applying the quadratic formula include:

• Identify coefficients \( a \), \( b \), and \( c \) from the quadratic equation.
• Substitute these into the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Solve for \( x \) to find the roots, which denote where the parabola intersects the x-axis.

Understanding and applying the quadratic formula is essential not just for solving equations but also for deeper insight into how quadratics behave. Solving the quadratic can sometimes even assist with verifying maximum and minimum values through graph intersection points.