Problem 12 - Technical Mathematics with Calculus

Question

$$y=24-x^{2} \quad \text { from } x=2 \text { to } 4$$

Step-by-Step Solution

Answer

The function takes the value 20 at x=2 and the value 8 at x=4.

Step 1: Understand the Function

The exercise gives a function, which is a parabola that opens downwards as the coefficient of the squared term is negative. The function is defined as y = 24 - x^2.

Step 2: Plot the Function (Optional)

It helps to sketch the graph of the parabola to understand its behavior from x = 2 to x = 4. This visualization is not necessary for the calculation but helps to conceptualize the problem.

Step 3: Evaluate the Function at x = 2

Substitute x = 2 into the function to find the corresponding value of y. y = 24 - (2)^2 = 24 - 4 = 20

Step 4: Evaluate the Function at x = 4

Substitute x = 4 into the function to find the corresponding value of y. y = 24 - (4)^2 = 24 - 16 = 8

Step 5: State the Values of the Function

The value of the function at x = 2 is y = 20, and at x = 4, the value of the function is y = 8.

Key Concepts

Quadratic Function

At the core of algebra is the concept of a quadratic function, which is a type of polynomial equation represented by the general form \( y = ax^2 + bx + c \). When we look at \( y = 24 - x^2 \) from our exercise, it's a quadratic function because it involves \( x^2 \) as the highest power of the variable \( x \).

Key features of a quadratic function include its shape, which is a parabola, and its direction, which can open upwards or downwards. This direction depends on the coefficient of the \( x^2 \) term. In our case, since the coefficient is \( -1 \) (implied from \( -x^2 \)), the parabola opens downwards. This tells us that the graph has a maximum point - the vertex of the parabola - and then extends towards negative infinity on both sides as \( x \) moves away from the vertex.

Understanding the structure and behavior of quadratic functions is essential for solving equations and predicting the shape and direction of their graphs without even plotting them on the coordinate plane.

Parabola Graphing

The process of parabola graphing involves plotting a curve based on a quadratic function. The graph of a quadratic function is always a parabola, which is a symmetric curve that can either be concave up (opening upwards) or concave down (opening downwards).

When graphing \( y = 24 - x^2 \) within a coordinate plane, you would find that the vertex, the highest point on the curve since the parabola opens downwards, is at the point (0,24). From there, the value of \( y \) decreases as \( x \) moves away from zero.

Plotting the function from \( x=2 \) to \( x=4 \) helps visualize how the function decreases. You'd start at the point (2,20) and move to (4,8). Drawing these points and connecting them smoothly reveals the shape of the parabola between these two \( x \) values. This visual tool is incredibly helpful for understanding the function's behavior over a specific interval and for finding the maximum or minimum values, intercepts, and intervals of increase or decrease.

Function Evaluation

Evaluating functions means calculating the output for specific inputs. For quadratic functions, this involves substituting the \( x \) value into the function to find the corresponding \( y \) value.

The steps to evaluate \( y = 24 - x^2 \) at \( x=2 \) and \( x=4 \) were clearly demonstrated: \( y = 24 - (2)^2 = 20 \) and \( y = 24 - (4)^2 = 8 \) respectively. Each evaluation gave us the specific \( y \) value at these points, illustrating how the function's value changes with different \( x \) values.

Practicing function evaluation helps in understanding not just single points, but also the overall behavior of a function across an interval or its entire domain. Furthermore, mastering the evaluation technique provides a crucial foundation for more complex mathematical problems such as calculus, where evaluating functions becomes a frequent and essential task.