Finding 'a' In A Parabola Equation With Zeros 6 And -5

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The study of parabolas is a fundamental aspect of algebra and calculus. Parabolas are U-shaped curves that can be described by quadratic equations. Understanding their properties, such as zeros (roots or x-intercepts) and the vertex, is crucial in various applications, from physics to engineering. This article delves into a specific problem involving a parabola with given zeros and a point on its graph. We aim to determine the equation that can be solved to find the value of the leading coefficient, denoted as 'a', in the quadratic equation. Finding the equation of a parabola given certain conditions is a common task in mathematics, and this article will provide a comprehensive explanation of the process.

Understanding Parabolas and Quadratic Equations

A parabola is the graphical representation of a quadratic equation, which has the general form:

f(x)=ax2+bx+cf(x) = ax^2 + bx + c

Where:

  • f(x) represents the y-coordinate of the parabola.
  • x is the independent variable.
  • a, b, and c are constants, with a not equal to zero.

The value of 'a' determines the direction and width of the parabola. If a > 0, the parabola opens upwards, and if a < 0, it opens downwards. The larger the absolute value of a, the narrower the parabola. The zeros of a parabola are the x-values where the parabola intersects the x-axis, i.e., where f(x) = 0. These zeros are also known as the roots or x-intercepts of the quadratic equation. A parabola can have two distinct real zeros, one repeated real zero, or no real zeros (in which case it has complex zeros). Knowing the zeros of a parabola is a key step in determining its equation. The zeros provide us with factors of the quadratic equation, which helps in constructing the equation in factored form. Understanding quadratic equations is essential for solving various mathematical problems, and parabolas provide a visual representation of these equations.

Forms of a Quadratic Equation

Quadratic equations can be expressed in three main forms, each offering different insights into the parabola's properties:

  1. Standard Form: f(x) = ax^2 + bx + c
    • This form is useful for identifying the coefficients a, b, and c, which are used in the quadratic formula and other calculations.
  2. Vertex Form: f(x) = a(x - h)^2 + k
    • The vertex form directly reveals the vertex of the parabola, which is the point (h, k). The vertex is the minimum or maximum point of the parabola, depending on the sign of a.
  3. Factored Form: f(x) = a(x - r_1)(x - r_2)
    • The factored form is particularly useful when the zeros r_1 and r_2 are known. This form allows for easy identification of the x-intercepts of the parabola.

The Significance of Zeros

The zeros of a parabola, also known as roots or x-intercepts, are the points where the parabola intersects the x-axis. These points are crucial because they provide key information about the quadratic equation. If we know the zeros of a parabola, we can express the quadratic equation in its factored form. Zeros are significant because they help in understanding the behavior and properties of the parabola. For example, the zeros can be used to find the axis of symmetry, which is the vertical line that passes through the vertex of the parabola. The axis of symmetry is equidistant from the zeros, and its equation is given by:

x=r1+r22x = \frac{r_1 + r_2}{2}

Where r_1 and r_2 are the zeros of the parabola. The vertex lies on the axis of symmetry, so knowing the zeros helps in finding the x-coordinate of the vertex. The y-coordinate of the vertex can then be found by substituting this x-coordinate into the equation of the parabola. The zeros also play a crucial role in solving quadratic inequalities. By finding the zeros, we can determine the intervals where the parabola is above or below the x-axis, which helps in solving inequalities of the form ax^2 + bx + c > 0 or ax^2 + bx + c < 0.

Problem Statement: Finding the Equation

Consider a parabola with zeros at x = 6 and x = -5. We are also given a point on the graph of the parabola, (-1, 3). The goal is to find the equation that can be solved to determine the value of a in the equation of the parabola. This problem highlights the importance of using given information to construct the equation of a parabola. The zeros provide us with the factors of the quadratic equation, and the given point allows us to solve for the leading coefficient a. Solving for 'a' is a crucial step in determining the unique equation of the parabola that satisfies the given conditions.

Utilizing the Factored Form

Since we know the zeros of the parabola, x = 6 and x = -5, we can express the quadratic equation in factored form. The factored form of a quadratic equation is given by:

f(x)=a(x−r1)(x−r2)f(x) = a(x - r_1)(x - r_2)

Where r_1 and r_2 are the zeros of the parabola. In this case, r_1 = 6 and r_2 = -5. Substituting these values into the factored form, we get:

f(x)=a(x−6)(x−(−5))f(x) = a(x - 6)(x - (-5))

Simplifying, we have:

f(x)=a(x−6)(x+5)f(x) = a(x - 6)(x + 5)

This equation represents the parabola with the given zeros. However, we still need to find the value of a. To do this, we will use the given point on the graph, (-1, 3). Factored form is a powerful tool for expressing quadratic equations when the zeros are known. It simplifies the process of constructing the equation and allows for easy substitution of given points to solve for unknown coefficients.

Using the Given Point to Solve for 'a'

We are given that the point (-1, 3) lies on the graph of the parabola. This means that when x = -1, the value of f(x) is 3. We can substitute these values into the equation we derived in the previous section:

f(x)=a(x−6)(x+5)f(x) = a(x - 6)(x + 5)

Substituting x = -1 and f(x) = 3, we get:

3=a(−1−6)(−1+5)3 = a(-1 - 6)(-1 + 5)

This is the equation we need to solve for a. The equation represents the condition that the point (-1, 3) must satisfy for the parabola with zeros at 6 and -5. Solving for 'a' involves substituting the coordinates of the given point into the factored form of the quadratic equation and then simplifying the resulting expression.

Analyzing the Options

Now, let's analyze the given options and determine which one matches the equation we derived:

A. 3 = a(-1 + 6)(-1 - 5) B. 3 = a(-1 - 6)(-1 + 5) C. -1 = a(3 + 6)(3 - 5) D. -1 = a(3 - 6)(3 + 5)

Comparing these options with the equation we derived:

3=a(−1−6)(−1+5)3 = a(-1 - 6)(-1 + 5)

We can see that option B matches our equation exactly. The other options do not correctly substitute the given point and zeros into the factored form of the quadratic equation. Option A has incorrect signs within the parentheses, while options C and D incorrectly swap the x and y coordinates of the given point.

Detailed Explanation of Option B

Option B, 3 = a(-1 - 6)(-1 + 5), is the correct equation to solve for a. Let's break down why this equation is correct:

  • The left side of the equation, 3, represents the y-coordinate of the given point (-1, 3). This is the value of f(x) when x = -1.
  • The right side of the equation, a(-1 - 6)(-1 + 5), represents the factored form of the quadratic equation evaluated at x = -1.
    • The term (-1 - 6) represents (x - 6) with x = -1.
    • The term (-1 + 5) represents (x + 5) with x = -1.
    • The coefficient a is the leading coefficient of the quadratic equation, which we are trying to find.

This equation correctly sets the y-coordinate of the given point equal to the value of the quadratic equation at that point, allowing us to solve for a. Option B is the only option that accurately represents the substitution of the given point and zeros into the factored form of the quadratic equation.

Conclusion

In this article, we explored the process of finding the equation that can be solved to determine the value of a in the equation of a parabola, given its zeros and a point on the graph. We began by understanding the fundamentals of parabolas and quadratic equations, including the standard, vertex, and factored forms. We then used the given zeros to express the quadratic equation in factored form and substituted the coordinates of the given point to create an equation that can be solved for a. In conclusion, understanding the properties of parabolas and quadratic equations is essential for solving a variety of mathematical problems. By using the factored form and substituting given points, we can effectively determine the equation of a parabola and the value of its leading coefficient.

The correct equation to solve for a is:

3=a(−1−6)(−1+5)3 = a(-1 - 6)(-1 + 5)

This corresponds to option B in the given choices.