Finding The Quadratic Function Equation With Vertex (2,-25) And X-intercept (7,0)
In the realm of mathematics, quadratic functions hold a prominent position, serving as fundamental building blocks in various applications. These functions, characterized by their parabolic curves, are defined by a standard equation that encapsulates their essential properties. One of the key challenges in working with quadratic functions lies in determining their specific equations given certain characteristics, such as the vertex and x-intercepts.
Decoding the Vertex Form of a Quadratic Equation
The vertex form of a quadratic equation provides a powerful tool for unraveling the equation when the vertex and another point on the parabola are known. This form, expressed as:
f(x) = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola and 'a' dictates the parabola's stretch or compression and direction of opening. In essence, the vertex form elegantly encapsulates the parabola's turning point and its overall shape.
Leveraging the Given Vertex
In our specific scenario, we are presented with a quadratic function whose vertex resides at the coordinates (2, -25). This crucial piece of information allows us to directly substitute these values into the vertex form equation. By replacing 'h' with 2 and 'k' with -25, we arrive at a modified equation:
f(x) = a(x - 2)^2 - 25
This equation now incorporates the vertex, but the coefficient 'a' remains an unknown, preventing us from fully defining the quadratic function.
Incorporating the x-intercept
To bridge the gap and determine the value of 'a', we turn to the provided x-intercept, which is located at (7, 0). An x-intercept, by definition, is a point where the parabola intersects the x-axis, implying that the function's value (f(x)) is zero at this point. By substituting x = 7 and f(x) = 0 into our modified equation, we introduce a new equation:
0 = a(7 - 2)^2 - 25
This equation now features 'a' as the sole unknown, making it solvable through algebraic manipulation.
Solving for the Stretch Factor 'a'
To isolate 'a', we embark on a series of steps. First, we simplify the equation by evaluating the expression within the parentheses:
0 = a(5)^2 - 25
Next, we further simplify by squaring 5:
0 = 25a - 25
To isolate the term containing 'a', we add 25 to both sides of the equation:
25 = 25a
Finally, we divide both sides by 25 to solve for 'a':
a = 1
With the value of 'a' determined, we now possess all the necessary components to construct the complete quadratic equation.
Constructing the Quadratic Equation
By substituting the calculated value of 'a' (1) back into the vertex form equation, we arrive at the final equation of the quadratic function:
f(x) = 1(x - 2)^2 - 25
Simplifying this equation, we obtain:
f(x) = (x - 2)^2 - 25
Expanding to Standard Form (Optional)
While the vertex form provides a concise representation of the quadratic function, it is often beneficial to expand it into the standard form, which is expressed as:
f(x) = ax^2 + bx + c
Expanding our equation, we first address the squared term:
f(x) = (x^2 - 4x + 4) - 25
Then, we combine the constant terms:
f(x) = x^2 - 4x - 21
This standard form representation provides an alternative perspective on the quadratic function, highlighting its coefficients and constant term.
Validating the Solution
To ensure the accuracy of our derived equation, it is prudent to verify that it satisfies the given conditions. We can achieve this by substituting the x-intercept (7, 0) into the equation:
f(7) = (7 - 2)^2 - 25
Simplifying, we get:
f(7) = (5)^2 - 25
Further simplification yields:
f(7) = 25 - 25 = 0
This confirms that the equation correctly predicts the function's value at the x-intercept.
Factoring the Quadratic Equation
Another valuable technique for analyzing quadratic equations is factoring. Factoring allows us to express the quadratic as a product of two linear expressions, which can reveal the x-intercepts of the parabola. Factoring our standard form equation:
f(x) = x^2 - 4x - 21
we seek two numbers that multiply to -21 and add to -4. These numbers are -7 and 3, allowing us to factor the equation as:
f(x) = (x - 7)(x + 3)
This factored form directly reveals the x-intercepts at x = 7 and x = -3. The x-intercept at x = 7 aligns with the provided information, while the x-intercept at x = -3 provides an additional characteristic of the parabola.
Identifying the Correct Option
Comparing our derived equations with the provided options, we find that option (x - 7)(x + 3) matches our factored form, confirming it as the correct answer. This option accurately represents the quadratic function with the specified vertex and x-intercept.
Visualizing the Parabola (Optional)
To further solidify our understanding, it can be insightful to visualize the parabola represented by the equation. The vertex form readily provides the vertex coordinates (2, -25), while the factored form reveals the x-intercepts at x = 7 and x = -3. With these key points, we can sketch the parabola, confirming its shape and position on the coordinate plane. The parabola opens upwards due to the positive coefficient of the x^2 term in the standard form.
Exploring Alternative Forms (Optional)
While the vertex and factored forms have proven instrumental in solving this problem, other forms of quadratic equations exist, each offering unique advantages. The general form, expressed as:
f(x) = ax^2 + bx + c
provides a comprehensive representation of the quadratic function, encompassing all its coefficients. Converting between these forms can offer a deeper understanding of the relationship between the equation's parameters and the parabola's characteristics.
Conclusion
In summary, we have successfully determined the equation of the quadratic function with a vertex at (2, -25) and an x-intercept at (7, 0) by leveraging the vertex form of the quadratic equation, incorporating the given x-intercept, and solving for the stretch factor 'a'. Through algebraic manipulation, we arrived at the equation f(x) = (x - 2)^2 - 25, which can be further expressed in standard form as f(x) = x^2 - 4x - 21 and factored form as f(x) = (x - 7)(x + 3). This solution aligns with option (x - 7)(x + 3). This exploration highlights the power of quadratic equations and the diverse techniques available for their analysis. Understanding these concepts forms a solid foundation for further mathematical endeavors.
This comprehensive approach not only solves the problem but also reinforces key concepts related to quadratic functions, their properties, and their various representations. By mastering these concepts, you can confidently tackle a wide range of mathematical challenges.
