Problem 31 - Applied Calculus

Question

For each quadratic function: a. Find the vertex using the vertex formula. b. Graph the function on an appropriate window. (Answers may differ.) $$ f(x)=x^{2}-40 x+500 $$

Step-by-Step Solution

Answer

Vertex is (20, 100); use appropriate window to graph the parabola.

Step 1: Identify the Quadratic Function Coefficients

In the quadratic function \( f(x) = ax^2 + bx + c \), identify the coefficients: \( a = 1 \), \( b = -40 \), and \( c = 500 \).

Step 2: Use the Vertex Formula for x-coordinate

The x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \). Substitute \( b = -40 \) and \( a = 1 \) into the formula: \( x = -\frac{-40}{2 \times 1} = \frac{40}{2} = 20 \).

Step 3: Calculate the y-coordinate of the Vertex

Substitute the x-coordinate of the vertex back into the function to find the y-coordinate: \( f(20) = (20)^2 - 40 \times 20 + 500 = 400 - 800 + 500 = 100 \).

Step 4: Verify the Vertex Coordinates

Thus, the vertex of the function is \((20, 100)\). Check the calculations to ensure no errors were made in previous steps.

Step 5: Graph the Quadratic Function

Plot the vertex \((20, 100)\) on a graph. Identify the axis of symmetry which is the line \( x = 20 \). The parabola opens upwards because \( a = 1 > 0 \). Plot additional points to ensure the curve's correct shape, based on transformations from the parent function \( y = x^2 \).

Step 6: Set an Appropriate Window for Graphing

Choose a graphing window that includes the vertex \((20, 100)\) and ensures the parabola shape is clear. For accuracy, consider an x-axis range from 10 to 30 and a y-axis range from 0 to 200.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool to solve quadratic equations of the form \( ax^2 + bx + c = 0 \). It is particularly useful when factoring is difficult or impossible. The formula is given by\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where:

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term

By substituting these coefficients into the formula, you can find the solutions to the quadratic equation. These solutions are the x-values where the graph of the quadratic crosses the x-axis.
It is important to note that not all quadratic equations will have real solutions, depending on the discriminant \( b^2 - 4ac \). If the discriminant is positive, there are two distinct real solutions. If it is zero, there is exactly one real solution. And if it is negative, there are no real solutions, only complex ones.

Graphing Quadratics

When graphing quadratic functions, the shape of the graph is always a parabola. This curve can open either upwards or downwards.
It opens upwards if the coefficient \( a \) is positive. Conversely, it opens downwards if \( a \) is negative. The general approach to graphing a quadratic function involves identifying its key features, including:

• The vertex, which is the highest or lowest point.
• The axis of symmetry, which divides the parabola into two mirror-image halves.
• Intercepts, where the graph crosses the axes.

For accurate graphing, one should first determine the vertex, then use additional points on either side of the vertex to confirm the parabola's symmetrical shape. It is helpful to set an appropriate viewing window that captures all significant aspects of the graph, ensuring the main parts of the function are visible and easy to understand.

Parabola Vertex

The vertex of a parabola is a pivotal point that represents either its minimum or maximum. It is found at the "turning point" of the graph. To find the vertex using the standard form \( f(x) = ax^2 + bx + c \), use the formula for the x-coordinate of the vertex: \( x = -\frac{b}{2a} \).
Once the x-coordinate is found, substitute it back into the original quadratic equation to calculate the corresponding y-coordinate. Together, these coordinates \((x, y)\) form the vertex of the parabola.
This point is essential because it determines the graph's peak or dip. For our example, with a quadratic function \( f(x) = x^2 - 40x + 500 \), the vertex was determined as \((20, 100)\). Knowing the vertex allows for more straightforward and more accurate graphing, as it provides a central reference point.

Axis of Symmetry

The axis of symmetry of a quadratic function is an imaginary vertical line that bisects the parabola into two symmetrical halves. The formula for finding the axis of symmetry in the quadratic equation \( f(x) = ax^2 + bx + c \) is \( x = -\frac{b}{2a} \), the same as the x-coordinate of the vertex.
This line is crucial because it reflects the natural balance of a quadratic graph. It ensures that for every point on one side, there is a corresponding point on the opposite side. In our specific example, the axis of symmetry is located at \( x = 20 \).
This concept simplifies the process of graphing quadratic functions by allowing you to plot points symmetrically, ensuring the graph reflects accurately. Understanding the axis of symmetry helps clarify how quadratic functions behave across their domain.