Exploring F(x) = 4x² - 2x + 2 Finding X-Intercepts And Vertex

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Hey guys! Today, we're diving deep into the fascinating world of quadratic functions, and we'll be dissecting the function f(x) = 4x² - 2x + 2. We're going to uncover its secrets, from finding its x-intercepts to pinpointing its vertex. So, buckle up and let's get started!

Cracking the Code: Finding the X-Intercepts

Let's kick things off by tackling the x-intercepts. Remember, the x-intercepts are those special points where the graph of our function crosses the x-axis. At these points, the value of f(x) is exactly zero. So, to find them, we need to solve the equation:

4x² - 2x + 2 = 0

Now, this looks like a job for the quadratic formula! For any quadratic equation in the form ax² + bx + c = 0, the solutions for x are given by:

x = (-b ± √(b² - 4ac)) / 2a

In our case, we have a = 4, b = -2, and c = 2. Let's plug these values into the formula and see what we get:

x = (2 ± √((-2)² - 4 * 4 * 2)) / (2 * 4) x = (2 ± √(4 - 32)) / 8 x = (2 ± √(-28)) / 8

Uh oh! We've encountered a negative number under the square root. This means that the solutions for x are complex numbers, not real numbers. In the context of graphing on the Cartesian plane (our usual x-y plane), this implies that the graph of our function never actually crosses the x-axis. Therefore, this quadratic equation has no real roots, meaning there are no x-intercepts for this function. So, when you're filling in the blanks, you'd select "None" – simple as that, guys!

Understanding the Discriminant

This brings us to a cool concept called the discriminant. The discriminant is the part of the quadratic formula under the square root, which is b² - 4ac. It tells us a lot about the nature of the roots (the solutions for x):

  • If the discriminant is positive (b² - 4ac > 0), the quadratic equation has two distinct real roots (two x-intercepts).
  • If the discriminant is zero (b² - 4ac = 0), the quadratic equation has one real root (one x-intercept, also known as a repeated root).
  • If the discriminant is negative (b² - 4ac < 0), the quadratic equation has no real roots (no x-intercepts), but it has two complex roots.

In our case, the discriminant is -28, which is negative. This confirms our finding that the function has no x-intercepts. Isn't math neat?

Finding the Vertex: The Peak or Valley of Our Parabola

Now that we've conquered the x-intercepts (or lack thereof!), let's move on to another crucial feature of our quadratic function: the vertex. The vertex is the point where the parabola (the U-shaped graph of a quadratic function) changes direction. It's either the highest point (the maximum) or the lowest point (the minimum) on the graph.

There are a couple of ways to find the vertex. One way is to use the vertex formula. For a quadratic function in the form f(x) = ax² + bx + c, the x-coordinate of the vertex is given by:

x_vertex = -b / 2a

And once we have the x-coordinate, we can plug it back into the function to find the y-coordinate of the vertex:

_y_vertex = f(x_vertex)

Let's apply this to our function, f(x) = 4x² - 2x + 2. We already know that a = 4 and b = -2. So, the x-coordinate of the vertex is:

x_vertex = -(-2) / (2 * 4) = 2 / 8 = 1/4 = 0.25

Now, let's plug this value back into the function to find the y-coordinate:

y_vertex = f(0.25) = 4(0.25)² - 2(0.25) + 2 y_vertex = 4(0.0625) - 0.5 + 2 y_vertex = 0.25 - 0.5 + 2 y_vertex = 1.75

Therefore, the vertex of our function is at the point (0.25, 1.75). So, that's what you'll put in those parentheses, guys!

Alternative Method: Completing the Square

There's another cool technique for finding the vertex called "completing the square." It involves rewriting the quadratic function in vertex form, which is:

f(x) = a(x - h)² + k

where (h, k) is the vertex of the parabola. Let's see how it works for our function:

f(x) = 4x² - 2x + 2

First, we factor out the coefficient of the x² term (which is 4) from the first two terms:

f(x) = 4(x² - (1/2)x) + 2

Now, we need to complete the square inside the parentheses. To do this, we take half of the coefficient of the x term (which is -1/2), square it ((-1/4)² = 1/16), and add and subtract it inside the parentheses:

f(x) = 4(x² - (1/2)x + 1/16 - 1/16) + 2

Now, we can rewrite the first three terms inside the parentheses as a squared term:

f(x) = 4((x - 1/4)²) - 4(1/16) + 2

Simplify:

f(x) = 4(x - 1/4)² - 1/4 + 2 f(x) = 4(x - 0.25)² + 1.75

Now, we're in vertex form! We can directly read off the vertex as (h, k) = (0.25, 1.75), which matches our previous result. Pretty slick, huh?

Putting It All Together: A Complete Picture of f(x)

So, we've explored our function, f(x) = 4x² - 2x + 2, from different angles. We discovered that it has no x-intercepts (it never crosses the x-axis) and that its vertex is at the point (0.25, 1.75). This tells us that the parabola opens upwards (since the coefficient of x² is positive) and its minimum point is at (0.25, 1.75). This function will always output positive values.

By finding these key features, we can start to sketch the graph of the function or, even better, use graphing software to visualize it perfectly. Understanding these concepts allows us to analyze and predict the behavior of quadratic functions in various real-world scenarios, from projectile motion to optimization problems.

Why This Matters: Real-World Applications

You might be wondering, "Okay, this is cool math stuff, but where does it actually matter?" Well, quadratic functions pop up all over the place in the real world!

  • Physics: The trajectory of a projectile (like a ball thrown in the air) follows a parabolic path, which can be modeled by a quadratic function. Knowing the vertex helps us determine the maximum height the projectile reaches.
  • Engineering: Engineers use quadratic functions to design arches, bridges, and other structures. The parabolic shape is often used for its strength and stability.
  • Business: Businesses use quadratic functions to model profit and cost curves. Finding the vertex helps them determine the optimal price point to maximize profit.
  • Computer Graphics: Quadratic functions are used to create smooth curves and shapes in computer graphics and animations.

So, as you can see, understanding quadratic functions isn't just about solving equations; it's about understanding the world around us!

Wrapping Up: The Power of Quadratic Functions

We've taken a deep dive into the quadratic function f(x) = 4x² - 2x + 2 today. We've learned how to find the x-intercepts (or determine when they don't exist), how to pinpoint the vertex using multiple methods, and how these concepts connect to real-world applications. Hopefully, you've gained a new appreciation for the power and versatility of quadratic functions, guys! Keep exploring, keep learning, and remember that math is everywhere!