Analyzing F(x) = 3x² + 7x + 2: Discriminant, Intercepts & Zeros
Hey guys, let's dive into the world of quadratic functions! Today, we're going to explore the function $f(x)=3 x^2+7 x+2$. We'll uncover its secrets by calculating the discriminant, determining the number of x-intercepts, and finding its zeros. So, buckle up, and let's get started! This is a super important concept in math, and understanding it will help you ace your algebra tests and even understand real-world problems. We'll break down each part step by step so that you can easily follow along. You will learn the basics of quadratic equations, including the discriminant, the number of x-intercepts, and how to find the zeros or roots of the equation. These concepts are not only important for math class but can also be applied in many different fields, such as physics, engineering, and economics. Learning about this stuff will make you a math rockstar! Also, we will ensure that the text is well-organized, and easy to understand. Are you ready?
Calculating the Discriminant of $f(x)=3 x^2+7 x+2$
Alright, first things first, let's figure out the discriminant of our function $f(x)=3 x^2+7 x+2$. The discriminant is a super useful tool that helps us understand the nature of the roots (or zeros) of a quadratic equation. It tells us whether the equation has two distinct real roots, one real root (a repeated root), or no real roots (two complex roots). Now, for a quadratic equation in the form of $ax^2 + bx + c = 0$, the discriminant is calculated using the formula: $Δ = b^2 - 4ac$. In our case, our function is $f(x)=3 x^2+7 x+2$, which can also be written as $3x^2 + 7x + 2 = 0$. So, we have: a = 3, b = 7, and c = 2. Plugging these values into the discriminant formula, we get: $Δ = (7)^2 - 4 * (3) * (2)$. Let's do the math! $Δ = 49 - 24$. Therefore, $Δ = 25$. Wow! The discriminant is 25. The result shows us that the discriminant is a positive number. This tells us that the quadratic equation has two distinct real roots. Now that we know the value of the discriminant, we can move on to find out how many x-intercepts the function has. The discriminant is a fundamental concept in algebra, allowing you to analyze the behavior of quadratic equations without actually solving them. Knowing the discriminant helps determine the nature of the roots, providing insights into the graph's behavior and how it interacts with the x-axis. Keep in mind, the value of the discriminant can quickly tell us a lot about the quadratic function. It gives us a quick way to check the roots of the quadratic equation. The positive discriminant means that the quadratic function will intersect the x-axis at two distinct points. And understanding all the pieces is super important.
Determining the Number of x-Intercepts
Alright, so now that we've crunched the numbers and found the discriminant, let's figure out how many x-intercepts our function has. The x-intercepts, also known as the zeros or roots, are the points where the graph of the function crosses the x-axis. Remember our discriminant, $Δ = 25$. Because the discriminant is positive, we know that the quadratic function will have two distinct real roots. This means the graph of the function will intersect the x-axis at two different points. Each x-intercept corresponds to a real solution to the quadratic equation $3x^2 + 7x + 2 = 0$. When the discriminant is positive, the quadratic equation has two real roots, meaning the parabola opens up, intersecting the x-axis twice. The number of x-intercepts tells us a lot about the graph of the function. Specifically, a positive discriminant indicates that there are two distinct x-intercepts. If the discriminant were zero, there would be one x-intercept. If it were negative, there would be no x-intercepts. Graphing the function will help you visualize the x-intercepts. In this case, since the discriminant is positive, we know the function will have two x-intercepts. You can always double-check this by graphing the function. It's also a good practice to use the quadratic formula to solve for the roots and confirm the x-intercepts. Doing this helps reinforce our understanding of the concepts. Also, it allows us to visually see the function and x-axis. We're gaining a solid understanding of quadratic functions. With practice, identifying the number of x-intercepts becomes super easy. Knowing this is a key step in analyzing the behavior of quadratic functions.
Finding the Zeros of the Function
Okay, time to find the zeros of our function $f(x)=3 x^2+7 x+2$. The zeros are the x-values where the function equals zero, i.e., where the graph intersects the x-axis. There are a couple of ways to find the zeros: factoring or using the quadratic formula. Let's use the quadratic formula. For a quadratic equation in the form $ax^2 + bx + c = 0$, the quadratic formula is: $x = rac-b ± \sqrt{b^2 - 4ac}}{2a}$. We already know that a = 3, b = 7, and c = 2. Substituting these values into the formula, we get}2 * 3}$. Let's simplify this}6}$. $x = rac{-7 ± \sqrt{25}}{6}$. And we know that $\sqrt{25} = 5$, so we get6}$. This gives us two possible solutions{6} = rac{-2}{6} = - rac{1}{3}$ and $x_2 = rac{-7 - 5}{6} = rac{-12}{6} = -2$. So, the zeros of the function are $x = - rac{1}{3}$ and $x = -2$. These are the x-intercepts we talked about earlier. These are the points where the graph of the function crosses the x-axis. We have successfully found the zeros of the quadratic equation! To find the zeros, we can either factor the quadratic equation or use the quadratic formula. Understanding how to find the zeros is super important for understanding the behavior of the quadratic function. These points are essential for sketching the graph. Also, they help us solve real-world problems. Knowing the x-intercepts allows us to see where the function crosses the x-axis, thus completing our analysis of $f(x)=3 x^2+7 x+2$. We used the quadratic formula. And now, we know exactly where the function crosses the x-axis, which means that we can understand how to graph the function with precision.
Summary
So, let's recap everything we've learned about the quadratic function $f(x)=3 x^2+7 x+2$:
- Discriminant: We calculated the discriminant and found that $Δ = 25$. This positive value tells us that the function has two distinct real roots.
- Number of x-intercepts: Because the discriminant is positive, the function has two x-intercepts.
- Zeros: We used the quadratic formula to find the zeros of the function, which are $x = - rac{1}{3}$ and $x = -2$. These are the x-values where the function crosses the x-axis.
This comprehensive analysis allows us to completely understand the function, its behavior, and its relationship to the x-axis. Quadratic functions are a fundamental concept in algebra, and understanding them will unlock a lot of doors for you. Understanding quadratic functions can open doors to advanced math and real-world applications. We have covered the essentials, including the discriminant, x-intercepts, and zeros. Keep practicing and applying what you've learned. It will help you master algebra, which is an essential skill. You will be well on your way to conquering quadratic functions!