Problem 31 - Brief Applied Calculus

Question

31-34. 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

The vertex is (20, 100), and the parabola opens upwards.

Step 1: Identify a, b, and c in the quadratic equation

The given quadratic function is \( f(x) = x^2 - 40x + 500 \). In a quadratic equation of the form \( ax^2 + bx + c \), the coefficients \( a \), \( b \), and \( c \) are the numbers in front of \( x^2 \), \( x \), and the constant term, respectively. Here, \( a = 1 \), \( b = -40 \), and \( c = 500 \).

Step 2: Use the vertex formula to find the vertex

The vertex \((h, k)\) of a quadratic function can be found using the formulas: \( h = -\frac{b}{2a} \) and \( k = f(h) \). Here, \( a = 1 \) and \( b = -40 \). First, calculate \( h = -\frac{-40}{2 \cdot 1} = 20 \). To find \( k \), substitute \( h = 20 \) back into the function: \( k = f(20) = 20^2 - 40 \cdot 20 + 500 = 400 - 800 + 500 = 100 \). Thus, the vertex is \((20, 100)\).

Step 3: Graph the quadratic function

To graph the function \( f(x) = x^2 - 40x + 500 \), plot the vertex (20, 100) on a coordinate plane. Since the parabola opens upwards (because the coefficient of \( x^2 \) is positive), sketch the curve such that it is symmetric about the vertical line \( x = 20 \). Select an appropriate range for the \( y \)-axis that includes the vertex and other points of interest, and use a range for the \( x \)-axis around the vertex to show the shape of the parabola.

Key Concepts

Vertex Formula

When working with quadratic functions, the vertex provides crucial information about the parabola's peak or valley. For a quadratic function in standard form, like \( f(x) = ax^2 + bx + c \), you can find the vertex using the vertex formula. The formula to determine the vertex \((h, k)\) is:

• Calculate \( h \) using \( h = -\frac{b}{2a} \).
• Then, compute \( k \) with \( k = f(h) \), by substituting \( h \) back into the function.

In our example, where \( a = 1 \), \( b = -40 \), the calculation for \( h \) becomes \( -\frac{-40}{2 \cdot 1} = 20 \). We then substitute \( h = 20 \) back into the function to find \( k \). Thus, \( k = f(20) = 100 \), which makes the vertex \((20, 100)\). Understanding how to use the vertex formula allows you to quickly find this fixed point, which is key in graphing and analysis.

Graphing Parabolas

Graphing a parabola involves plotting points such that the shape of the graph comes alive as a symmetric, U-shaped curve. Here's how to effectively graph the function \( f(x) = ax^2 + bx + c \):

• First, find the vertex \((h, k)\), as described using the vertex formula.
• Once you have the vertex, plot it on the coordinate plane.
• Decide the direction of opening for the parabola. In our quadratic \( f(x) = x^2 - 40x + 500 \), because \( a = 1 > 0 \), the parabola opens upwards.
• Ensure the graph is symmetrical around the vertical line passing through the vertex, here \( x = 20 \).
• Select an appropriate range for both \( x \) and \( y \) axes, ensuring it includes the vertex and stretches enough for a clear view of the parabola's shape.

Graphing is pivotal in understanding the behavior of quadratic functions, providing visual insights that equations alone do not articulate.

Quadratic Equations

Quadratic equations form the backbone of many ideas in algebra and are typically expressed in the format \( ax^2 + bx + c = 0 \). To solve these equations, one might:

• Factorize the quadratic if possible, breaking it into simpler linear equations.
• Apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) when factorization isn't feasible.
• Determine the discriminant \( D \), which is \( b^2 - 4ac \), to ascertain the nature of the roots:
• If \( D > 0 \), the equation has two distinct real roots.
• If \( D = 0 \), there is exactly one real root, which coincides with the vertex.
• If \( D < 0 \), the roots are complex and not visible on the real number graph.

The equation \( f(x) = x^2 - 40x + 500 \) can similarly be scrutinized or solved using such methods, giving more insight into the function's behavior across different domains. Quadratic equations offer a multitude of solutions and graphical interpretations that underline their importance in mathematics.