Problem 33 - Applied Calculus

Question

Find constants \(a\) and \(b\) so that the minimum for the parabola \(f(x)=x^{2}+a x+b\) is at the given point. $$(-2,-3)$$

Step-by-Step Solution

Answer

The constants are \( a = 4 \) and \( b = 1 \).

Step 1: Understand the Vertex Form

The vertex form of a parabola is given by the equation \( f(x) = (x - h)^2 + k \), where \((h, k)\) is the vertex or the minimum point. Here, the minimum point given is \((-2, -3)\).

Step 2: Translate the Vertex Form to Standard Form

To rewrite \( f(x) = (x + 2)^2 - 3 \) (since \( h = -2 \) and \( k = -3 \)) into standard form: \[ f(x) = (x + 2)^2 - 3 = x^2 + 4x + 4 - 3 = x^2 + 4x + 1. \]

Step 3: Compare with Standard Form

The standard form of the parabola is \( f(x) = x^{2} + ax + b \). By comparing \( x^2 + 4x + 1 \) with \( x^2 + ax + b \), we identify that \( a = 4 \) and \( b = 1 \).

Key Concepts

parabola equation

A parabola equation is a mathematical representation of a curved line that is shaped like an arch. When graphed, it forms a symmetric U-shaped curve. Parabolas are key concepts in algebra and calculus because they represent quadratic functions, which are equations in the form of \( f(x) = ax^2 + bx + c \). These equations allow us to model various real-world scenarios, such as the trajectory of a projectile or the path of light reflected off a surface.

Every parabola has an axis of symmetry, a point called the vertex, and possibly a minimum or maximum point depending on the direction of its opening. If the coefficient \( a \) is positive, the parabola opens upwards, and the vertex is a minimum point. Conversely, if \( a \) is negative, the parabola opens downwards, and the vertex is a maximum point. Understanding these foundational elements of a parabola gives insight into analyzing the graph and properties of quadratic equations.

standard form

The standard form of a parabola equation is expressed as \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. This form is incredibly useful because it provides a straightforward method to recognize the key properties and behaviors of a parabola.

• The coefficient \( a \) influences the direction and width of the parabola. A larger \( |a| \) means a narrower parabola, whereas a smaller \( |a| \) results in a wider one.
• The coefficient \( b \) affects the slope of the line of symmetry, impacting where the vertex appears on the graph.
• The constant \( c \) represents the y-intercept, or the point where the graph crosses the y-axis, at \( (0, c) \).

Rewriting equations into standard form simplifies comparison and calculation, as demonstrated in the exercise solution we reviewed. By identifying values of \( a \) and \( b \), such as \( a = 4 \) and \( b = 1 \), the equation becomes easier to solve and interpret.

minimum point

The minimum point of a parabola is the lowest point on its graph when it opens upwards. Mathematically, this is known as the vertex in situations where the parabola represents a minimum. Understanding how to find and use the minimum point is essential for solving optimization problems and analyzing quadratic functions.

• The minimum point is most easily identified in vertex form \( f(x) = (x-h)^2 + k \), where \( (h, k) \) is the vertex. The given exercise shows how to translate from vertex form \( (x+2)^2 - 3 \) to standard form \( x^2 + 4x + 1 \), ensuring the vertex remains at \( (-2, -3) \).
• To find minimum points directly from standard form, you use the formula \( h = -\frac{b}{2a} \) to determine the x-coordinate of the vertex, which helps in planning further analysis or solving real-world problems.
• In practical terms, the minimum point can signify optimal points in various scenarios, such as minimum cost or time, giving strategic value in fields beyond mathematics.