Problem 33 - Brief 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 (40, -3400). The graph is a downward-opening parabola.

Step 1: Identify Coefficients

Identify the coefficients from the quadratic function in standard form, which is given by \( f(x) = ax^2 + bx + c \). In this case, we have \( a = -1 \), \( b = -80 \), and \( c = -1800 \).

Step 2: Use the Vertex Formula

The vertex \((h, k)\) of a quadratic function can be found using the formula \( h = -\frac{b}{2a} \) and \( k = f(h) \). First calculate \( h \) using \( h = -\frac{-80}{2(-1)} = 40 \).

Step 3: Calculate the Vertex's Y-Coordinate

Substitute \( h = 40 \) back into the function to find \( k \). This is done by calculating \( f(40) = -40^2 - 80(40) - 1800 \). Simplify to find \( k = -3400 \).

Step 4: Write Down the Vertex

Now that you have both coordinates, the vertex of the function is \((40, -3400)\).

Step 5: Choose a Graphing Window

Select a graph window that accommodates the vertex and any additional points you wish to plot. For this function, you might choose the x-axis from -20 to 100 and the y-axis from -4000 to 0.

Step 6: Graph the Quadratic Function

Plot the function on the chosen graph window. Mark the vertex at \((40, -3400)\) and sketch the parabola opening downwards since \( a = -1 \) is negative.

Key Concepts

Vertex Formula

The vertex formula is a crucial tool in graphing and analyzing quadratic functions. The vertex of a quadratic function represents the highest or lowest point of the graph, depending on the direction the parabola opens. For a quadratic function in standard form, given by \( f(x) = ax^2 + bx + c \), the vertex \((h, k)\) can be calculated using the formulas:

• \( h = -\frac{b}{2a} \)
• \( k = f(h) \)

The formula \( h = -\frac{b}{2a} \) helps to find the \( x \)-coordinate of the vertex quickly. This is because it uses the coefficients \( a \) and \( b \) from the quadratic function. Once you have the value for \( h \), you substitute it back into the original function to find \( k \), which is the \( y \)-coordinate. This calculated point \((h, k)\) is the vertex of the parabola. For example, using the function \( f(x) = -x^2 - 80x - 1800 \), we found \( h = 40 \) and \( k = -3400 \), resulting in the vertex \((40, -3400)\). This step is essential since the vertex helps to graph the function correctly, showing the parabola's turning point.

Graphing Quadratic Equations

Graphing quadratic equations involves plotting the curve known as a parabola on a coordinate plane. This provides a visual representation of the equation and reveals important features like the vertex and direction of the opening. Quadratic functions are generally expressed in the form \( f(x) = ax^2 + bx + c \). When graphing, the vertex plays a central role since it is the peak or the lowest point.
The steps typically include:

• Identifying the vertex using the vertex formula.
• Choosing a graphing window that includes the vertex and a reasonable span of \( x \)-values before and after the vertex.
• Plotting the vertex and additional points on either side to capture the shape of the parabola.
• Sketching the parabola through these points.

For the function given, \( f(x) = -x^2-80x-1800 \), a graphing window around \( x = -20 \) to 100 and \( y = -4000 \) to 0 adequately covers the shape of the parabola, emphasizing the vertex at \((40, -3400)\). The parabola opens downward, which is consistent with a negative \( a \)-value, showcasing how the graph can capture the behavior and turning points of the equation.

Parabola

A parabola is the U-shaped graph that represents the set of points equidistant from a fixed point called the focus and a line called the directrix. In quadratic functions, parabolas are described by the equation \( f(x) = ax^2 + bx + c \). The vertex acts as either the peak or lowest point on the graph. This point is significant because it gives us information about the minimum or maximum value of the quadratic function.
Parabolas open either upwards or downwards based on the sign of the coefficient \( a \):

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), it opens downwards, like in our example with \( f(x) = -x^2 - 80x - 1800 \).

The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into mirror-like halves. This is defined by \( x = h \), where \( h \) is the \( x \)-coordinate of the vertex.
Understanding the structure and properties of parabolas helps in comprehending how quadratic functions behave. They show the possibilities for maximum height or minimum depth and can be applied to areas such as projectile motion and optimization problems in physics and economics.