Problem 41 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the standard form of the quadratic function that has the indicated vertex and whose graph passes through the given point. Use a graphing utility to verify your result. Vertex: (1,-2)\(; \quad\) Point: (-1,14)

Step-by-Step Solution

Answer

The standard form of the quadratic function that has the vertex (1, -2) and passes through the point (-1, 14) is \[ f(x) = 4x^2 - 8x + 2 \].

Step 1: Substitute Vertex and Point in the Vertex Form

Substitute the given vertex (1, -2) and point (-1, 14) into the equation \[ f(x) = a(x - h)^2 + k \], giving \[ 14 = a(-1 - 1)^2 - 2 \] which simplifies to \[ 14 = a(4) - 2 \] then solve for 'a', giving \[ a = 4 \].

Step 2: Rewrite the Function in Standard Form

Rewrite the function, \[ f(x) = a(x - h)^2 + k \], in the standard form using the calculated 'a' value and the vertex point. This gives \[ f(x) = 4(x - 1)^2 - 2 \]. Expanding and simplifying yields \[ f(x) = 4x^2 - 8x + 2\].

Step 3: Verify the result

Check if the found standard form quadratic function passes through the given point (-1, 14). By substituting 'x' with -1, we get \[ f(-1) = 4*(-1)^2 - 8*(-1) + 2 = 14 \]. This confirms that our function is correct, as it passes through the given point.

Key Concepts

Vertex of a Quadratic Function

The vertex of a quadratic function is a fundamental concept in understanding its graph. The vertex form of a quadratic function is given by the equation:
\[\begin{equation} f(x) = a(x - h)^2 + k \right), \right]\right) \end{equation}\] For example, when we are given a vertex at (h,k) = (1, -2), as in our exercise, we can directly substitute these values into the vertex form to start building our equation. But why is the vertex so critical? The vertex represents the highest or lowest point on the graph of a quadratic function, known as the peak or trough, respectively, and is crucial for understanding the function's shape and direction.
Knowing the vertex also helps us identify the line of symmetry for the function's graph, which is a vertical line running through the vertex. For our given example, the line of symmetry would be the vertical line x = 1. With these points in mind, finding the vertex is an essential first step in sketching the graph of any quadratic function.

Graphing Quadratic Functions

Graphing quadratic functions is a visual way to understand the behavior of these mathematical expressions. To graph a quadratic function, you need to identify several key features: the vertex, the axis of symmetry, the direction of opening, and the intercepts.

• Vertex: As discussed earlier, this point is at the peak or trough of the parabola, the graph's shape for a quadratic function.
• Axis of Symmetry: This is a vertical line that passes through the vertex. Every point on the graph has a mirror point on the opposite side of the axis.
• Direction of Opening: If 'a' (the coefficient before \[\begin{equation} \right) \end{equation}\]

In our example, the quadratic function has a positive 'a' value of 4, indicating the parabola opens upwards. This means our vertex point (1, -2) is the lowest point—the minimum—on the graph. Using this information along with additional points, like the one provided (-1, 14), we can draw an accurate graph of the quadratic function. A graphing utility can also help confirm our findings, ensuring we have plotted the function correctly.

Quadratic Function Applications

Quadratic functions have a wide range of applications in many fields such as physics, engineering, and economics. They are particularly useful in modeling situations where there is a rate of change that increases or decreases over time, and in analyzing the paths of projectiles.

• Projectile Motion: In physics, quadratic functions are used to model the parabolic path of objects under the influence of gravity in projectile motion, assuming no air resistance.
• Economics: Economists use quadratic functions to model cost, revenue, and profit functions to find the maximum profit or minimum cost conditions.
• Engineering: Engineers may apply quadratic functions when analyzing forces in a structure or designing parabolic shapes, such as satellite dishes, to focus signals.
• Statistics: Quadratics can define trend lines in data analysis, which are essential for predicting future data points or understanding data patterns.

Understanding how to work with quadratic functions, graph them, and interpret their vertex and intercepts allows us to solve real-world problems. The exercises of finding a standard form equation for a quadratic function by using its vertex and a point on its graph, as in our given exercise, is a basic skill that builds the foundation for these complex applications.