Problem 36 - Algebra and Trigonometry

Question

Find the maximum or minimum value of the function. $$ f(t)=10 t^{2}+40 t+113 $$

Step-by-Step Solution

Answer

The minimum value of the function is 73 at \( t = -2 \).

Step 1: Identify the type of quadratic function

The given function is quadratic, and it is in the form of \( f(t) = at^2 + bt + c \) where \( a = 10, b = 40, c = 113 \). In a quadratic function, if \( a > 0 \), the parabola opens upwards and has a minimum point, whereas if \( a < 0 \), it opens downwards and has a maximum point. Here, \( a = 10 > 0 \), so it has a minimum point.

Step 2: Calculate the vertex formula

The vertex of a quadratic function \( f(t) = at^2 + bt + c \) can be found using the vertex formula, \( t = \frac{-b}{2a} \). This value of \( t \) will give us either the maximum or minimum point of the parabola, which is called the vertex.

Step 3: Apply the vertex formula

Using the vertex formula, substitute \( b = 40 \) and \( a = 10 \) into the equation: \[ t = \frac{-40}{2 \times 10} = \frac{-40}{20} = -2 \]. So, the vertex occurs at \( t = -2 \).

Step 4: Find the minimum value of the function

Substitute \( t = -2 \) into the original function \( f(t) = 10t^2 + 40t + 113 \) to find the minimum value: \[ f(-2) = 10(-2)^2 + 40(-2) + 113 = 10(4) - 80 + 113 = 40 - 80 + 113 = 73 \].

Step 5: Conclusion of minimum value

The minimum value of the function \( f(t) = 10t^2 + 40t + 113 \) is \( 73 \), which occurs at \( t = -2 \).

Key Concepts

Vertex Formula

The vertex of a quadratic function is a crucial point that determines the maximum or minimum of the function. To find the vertex, we use the vertex formula, which is derived from the general quadratic equation in the form \( f(t) = at^2 + bt + c \). This formula is \( t = \frac{-b}{2a} \) and it helps locate the vertex by providing the \( t \) coordinate of this point.

In our example, because \( a = 10 \) and \( b = 40 \), applying the vertex formula gives us \( t = \frac{-40}{2 \times 10} = -2 \). Once we have the \( t \) value, we can substitute it back into the function to find the corresponding \( f(t) \) value. This pairing, \( (t, f(t)) \), is the vertex of the parabola.

Parabola

A parabola is a curve that graphically represents a quadratic function. It typically exhibits a U-shaped or inverted U-shape.

• When the coefficient \( a \) in the equation \( at^2 + bt + c \) is positive, the parabola opens upwards, resembling a smile.
• If \( a \) is negative, the parabola opens downwards, like a frown.

With \( a = 10 \) in our given function \( f(t) = 10t^2 + 40t + 113 \), the parabola opens upwards, confirming it will have a minimum point rather than a maximum. Understanding the shape and direction of a parabola helps us predict the behavior of the function and find its key features like the vertex.

Minimum Value

In a quadratic function, the minimum value is the lowest point on its graph when the parabola opens upwards. This minimum occurs at the vertex of the parabola.

Using the vertex formula, we determined the \( t \) value at which the vertex occurs in our example. Substituting this \( t = -2 \) into the original function yields the function's minimum value. Calculations show that \( f(-2) = 73 \), making 73 the minimum value of the function \( f(t) = 10t^2 + 40t + 113 \). Finding the minimum value is essential, especially in real-world contexts, where this might represent the lowest cost or the shortest time.

Quadratic Equation

A quadratic equation is a polynomial equation of degree 2. In standard form, it is expressed as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \).

Quadratic equations can be solved using various methods like factoring, completing the square, or utilizing the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Each method is useful under different circumstances. However, identifying the vertex and understanding the structure of the quadratic function allows us to find important points on the graph such as intercepts and the vertex point itself, as shown in the example given with \( f(t) = 10t^2 + 40t + 113 \). Recognizing the form and properties of quadratic functions is fundamental when working with various mathematical problems and applications.