Problem 33 - 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.) $$ f(x)=-x^{2}-80 x-1800 $$

Step-by-Step Solution

Answer

The vertex is at (40, -6600). The parabola opens downward.

Step 1: Form of Quadratic Function

The given quadratic function is:\[ f(x) = -x^2 - 80x - 1800 \] This is in the standard form of quadratic equation: \( ax^2 + bx + c \) where \( a = -1 \), \( b = -80 \), and \( c = -1800 \).

Step 2: Find the Vertex x-coordinate

To find the x-coordinate of the vertex, use the vertex formula \( x = -\frac{b}{2a} \). Substitute the values of \( a \) and \( b \):\[ x = -\frac{-80}{2(-1)} = 40 \] So, the x-coordinate of the vertex is 40.

Step 3: Find the Vertex y-coordinate

To find the y-coordinate of the vertex, substitute \( x = 40 \) into the original function \( f(x) = -x^2 - 80x - 1800 \):\[ f(40) = -40^2 - 80(40) - 1800 \]\[ f(40) = -1600 - 3200 - 1800 \]\[ f(40) = -6600 \] So, the y-coordinate of the vertex is -6600.

Step 4: Identify the Vertex

Combining the x and y coordinates, the vertex of the quadratic function is \((40, -6600)\).

Step 5: Graph the Function

To graph the function:1. Plot the vertex \((40, -6600)\).2. Note the parabola opens downwards (as \( a = -1 \) is negative).3. Add points on either side of the vertex, and sketch the curve.This provides the basic shape and position of the function on a graph.

Key Concepts

Vertex Formula

One of the most critical aspects of working with quadratic functions is finding the vertex, which provides valuable information about the graph of the function. In any quadratic function given by the equation \( ax^2 + bx + c \), the vertex can be easily located using the vertex formula. This formula helps find the x-coordinate of the vertex with the expression \( x = -\frac{b}{2a} \).

Here's a step-by-step breakdown:

• Identify the coefficients from the quadratic equation: \( a \), \( b \), and \( c \).
• Substitute these coefficients into the vertex formula to find the x-coordinate.
• In the exercise, \( a = -1 \) and \( b = -80 \). Plug these into the formula to get \( x = 40 \).

Once you've determined the x-coordinate, the y-coordinate can be found by substituting the x-value back into the original equation. For the given function, substituting \( x = 40 \) returns \( f(40) = -6600 \). Thus, the vertex of the quadratic function is at the point \((40, -6600)\). This process is essential for graphing and understanding the behavior of quadratic functions.

Parabola Graphing

Graphing a parabola involves plotting its key features on a coordinate plane, particularly its vertex, direction, and shape. The vertex, found through the vertex formula, gives us a starting point for sketching the parabolic curve.

Here is a simple way to graph a quadratic function:

• Begin by plotting the vertex, determined earlier as \((40, -6600)\).
• Note the direction the parabola opens. If the coefficient \( a \) from the quadratic function is negative, the parabola opens downward, as is the case here.
• Choose additional points on either side of the vertex to determine the width and steepness of the parabola.

Once these elements are plotted, you simply connect the points in a smooth curve to complete the graph. The vertex will be the highest or lowest point depending on whether the parabola opens upwards or downwards. In this exercise, since the parabola opens downward, the vertex is the highest point of the parabola.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is instrumental in identifying the elements needed to analyze and graph a quadratic function. The standard form is written as \( ax^2 + bx + c \). Each coefficient has its importance in determining various aspects of the parabola's appearance and location.

• \( a \): The coefficient of \( x^2 \) influences the direction and width of the parabola. A negative \( a \) value means it will open downwards, while a positive value means upwards.
• \( b \): The linear coefficient is part of the vertex formula, affecting the x-coordinate of the vertex.
• \( c \): This is the constant term, representing the y-intercept of the parabola.

When analyzing the quadratic equation \( f(x) = -x^2 - 80x - 1800 \), you can see:

• \( a = -1 \), indicating the parabola will open downwards.
• \( b = -80 \) helps find the x-coordinate of the vertex.
• \( c = -1800 \) provides the position where the parabola crosses the y-axis.

Understanding and applying these elements from the standard form allows for accurate graphing and insight into the structure of quadratic functions. This approach demystifies how changes in these coefficients affect the graph and its corresponding real-world interpretations.