Quadratic Equation: Vertex (2,-25) & X-intercept (7,0)

Hey guys! Let's dive into a super interesting problem: finding the equation of a quadratic function. This isn't just any function; it's one that has a vertex at the point (2, -25) and crosses the x-axis at the point (7, 0). Sounds like a puzzle, right? Well, let’s break it down together and make it crystal clear. We’ll explore how to approach this step-by-step, ensuring you not only get the answer but also understand the underlying concepts. So, buckle up, and let’s get started!

Understanding Quadratic Functions

Before we jump into solving the problem, let’s quickly recap what quadratic functions are all about. A quadratic function is basically a polynomial function of degree 2, meaning the highest power of the variable x is 2. The general form of a quadratic function is:

$f(x) = ax^2 + bx + c$

Where a, b, and c are constants, and a isn't zero (otherwise, it would be a linear function). The graph of a quadratic function is a parabola, which is a U-shaped curve. This shape can either open upwards or downwards, depending on the sign of a. If a is positive, the parabola opens upwards, and if a is negative, it opens downwards.

Key Features of a Parabola

To really nail this, let's talk about the key features of a parabola that are super relevant to our problem:

• Vertex: This is the highest or lowest point on the parabola. If the parabola opens upwards, the vertex is the minimum point; if it opens downwards, the vertex is the maximum point. The vertex form of a quadratic equation is super handy here:

  $f(x) = a(x - h)^2 + k$

  Where (h, k) are the coordinates of the vertex.

• X-intercepts: These are the points where the parabola crosses the x-axis. At these points, $f(x) = 0$. We also call these the roots or zeros of the quadratic function. Knowing the x-intercepts can help us write the quadratic function in factored form.

• Axis of Symmetry: This is a vertical line that passes through the vertex and divides the parabola into two symmetrical halves. The equation of the axis of symmetry is $x = h$, where h is the x-coordinate of the vertex.

Setting Up the Equation

Okay, now that we've refreshed our understanding of quadratic functions and parabolas, let's tackle our specific problem. We're given that the vertex is at (2, -25) and there's an x-intercept at (7, 0). This is gold! We can use this information to build our equation.

Using the Vertex Form

Since we know the vertex, the vertex form of the quadratic equation is our best friend here. Remember, the vertex form is:

$f(x) = a(x - h)^2 + k$

Where (h, k) is the vertex. In our case, h = 2 and k = -25. So, we can plug these values in:

$f(x) = a(x - 2)^2 - 25$

Notice we still have a to figure out. This is where the x-intercept comes into play!

Using the X-intercept

We know that the parabola passes through the point (7, 0). This means when $x = 7$, $f(x) = 0$. Let's substitute these values into our equation:

$0 = a(7 - 2)^2 - 25$

Now we have an equation with just one unknown, a. Let's solve for it:

$0 = a(5)^2 - 25$

$0 = 25a - 25$

$25a = 25$

$a = 1$

Awesome! We found that a = 1. Now we have all the pieces we need.

Building the Quadratic Function

We now know that a = 1, h = 2, and k = -25. Let's plug these values back into the vertex form of the quadratic equation:

$f(x) = 1(x - 2)^2 - 25$

This looks good, but let's expand it to get the standard form and see if we can relate it to the given options:

$f(x) = (x - 2)^2 - 25$

$f(x) = (x^2 - 4x + 4) - 25$

$f(x) = x^2 - 4x - 21$

Factoring the Quadratic

To match the answer choices, let’s try factoring this quadratic. We're looking for two numbers that multiply to -21 and add up to -4. Those numbers are -7 and 3. So, we can factor the quadratic as:

$f(x) = (x - 7)(x + 3)$

Matching the Answer Choices

Now, let's take a look at the answer choices provided:

A. $f(x) = (x - 2)(x - 7)$ B. $f(x) = (x + 2)(x + 7)$ C. $f(x) = (x - 3)(x + 7)$ D. $f(x) = (x + 3)(x - 7)$

Comparing our factored form $f(x) = (x - 7)(x + 3)$ with the options, we can see that option D, $f(x) = (x + 3)(x - 7)$, matches perfectly. It’s just a matter of rearranging the factors, which doesn’t change the function since multiplication is commutative.

Conclusion

So, there you have it! The equation of the quadratic function with a vertex at (2, -25) and an x-intercept at (7, 0) is $f(x) = (x + 3)(x - 7)$. We solved this by using the vertex form of a quadratic equation, substituting the vertex coordinates, and then using the x-intercept to find the value of the leading coefficient. Finally, we expanded and factored the equation to match the given options.

Remember, guys, the key to mastering these problems is understanding the fundamental concepts and knowing how to apply them. Keep practicing, and you'll become a quadratic equation whiz in no time! Understanding quadratic functions is crucial, as they appear in various areas of mathematics and real-world applications, such as physics, engineering, and economics. By grasping the concepts of the vertex, x-intercepts, and the different forms of quadratic equations, you'll be well-equipped to tackle a wide range of problems. Don't hesitate to revisit these steps and try similar problems to solidify your understanding. Happy solving!