Graphing Quadratic Functions Finding Roots, Vertex, And More

Let's dive into the world of quadratic functions! In this article, we're going to break down how to find the key features of a quadratic function, specifically using the example f(x) = 3x² + 13x - 10. We'll cover finding the roots (also known as x-intercepts or zeros), the vertex, the axis of symmetry, the y-intercept, and finally, how to use all of this to graph the function. So, grab your calculators and let's get started!

Understanding Quadratic Functions

Before we jump into the specifics of our example, let's quickly recap what quadratic functions are all about. A quadratic function is a polynomial function of the form f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The graph of a quadratic function is a parabola, a U-shaped curve. The parabola can open upwards (if 'a' is positive) or downwards (if 'a' is negative). Key features of a parabola include the vertex (the minimum or maximum point), the axis of symmetry (a vertical line that divides the parabola into two symmetrical halves), the roots (the points where the parabola intersects the x-axis), and the y-intercept (the point where the parabola intersects the y-axis). Mastering these elements is crucial for understanding and analyzing quadratic functions, and this knowledge forms the foundation for solving various mathematical problems and real-world applications. So, let's get this straight, guys: quadratic functions are more than just equations; they are powerful tools for modeling and understanding the world around us.

Now, to find the vertex, roots, axis of symmetry and y-intercept, we will go through each step one by one. The vertex, being the turning point of the parabola, is a critical feature. It represents either the minimum value (if the parabola opens upwards) or the maximum value (if the parabola opens downwards) of the function. The coordinates of the vertex can be found using the formula (-b/2a, f(-b/2a)). This formula is derived from the process of completing the square, which transforms the quadratic equation into a form that reveals the vertex directly. The x-coordinate of the vertex, -b/2a, also gives us the equation of the axis of symmetry, a vertical line that passes through the vertex and divides the parabola into two mirror-image halves. Understanding the vertex and axis of symmetry helps us visualize the parabola's orientation and position in the coordinate plane. For instance, if we know the vertex is in the first quadrant and the parabola opens upwards, we can infer that the function's values will increase as we move away from the vertex in either direction along the x-axis. This kind of reasoning is invaluable in solving optimization problems, where we aim to find the maximum or minimum value of a function subject to certain constraints.

The roots, also known as x-intercepts or zeros, are the points where the parabola intersects the x-axis. At these points, the function's value, f(x), is equal to zero. Finding the roots is equivalent to solving the quadratic equation ax² + bx + c = 0. There are several methods to find the roots, including factoring, using the quadratic formula, and completing the square. Each method has its advantages and disadvantages, and the choice of method often depends on the specific quadratic equation. Factoring is the most straightforward method when the quadratic expression can be easily factored. However, not all quadratic expressions are factorable, and in such cases, we can turn to the quadratic formula. The quadratic formula, x = [-b ± √(b² - 4ac)] / 2a, provides a universal solution for finding the roots of any quadratic equation. It involves substituting the coefficients a, b, and c into the formula and simplifying. The discriminant, b² - 4ac, within the quadratic formula, plays a crucial role in determining the nature of the roots. If the discriminant is positive, there are two distinct real roots, meaning the parabola intersects the x-axis at two different points. If the discriminant is zero, there is one real root (a repeated root), indicating the parabola touches the x-axis at its vertex. If the discriminant is negative, there are no real roots, implying the parabola does not intersect the x-axis. Understanding the discriminant allows us to predict the number and nature of the roots even before solving the quadratic equation. Completing the square is another powerful method for finding the roots, and it also provides a pathway to convert the quadratic equation into vertex form, which directly reveals the vertex coordinates. By mastering these root-finding methods, we gain a comprehensive toolkit for analyzing quadratic functions and their behavior. The roots are not just mathematical solutions; they often represent critical points in real-world applications. For example, in physics, the roots of a projectile motion equation might represent the points where the projectile hits the ground.

1. Finding the Roots (x-intercepts) for f(x) = 3x² + 13x - 10

To find the roots, we need to solve the equation 3x² + 13x - 10 = 0. There are a couple of ways we can do this: factoring or using the quadratic formula. Let's try factoring first, as it can be quicker if it works.

Factoring Method

We need to find two numbers that multiply to (3 * -10) = -30 and add up to 13. Those numbers are 15 and -2. Now we rewrite the middle term:

3x² + 15x - 2x - 10 = 0

Next, we factor by grouping:

3x(x + 5) - 2(x + 5) = 0

(3x - 2)(x + 5) = 0

Now we set each factor equal to zero:

3x - 2 = 0 or x + 5 = 0

Solving for x, we get:

x = 2/3 or x = -5

So, the roots (x-intercepts) are x = 2/3 and x = -5. This means the parabola crosses the x-axis at these two points. Now, let's look at quadratic formula to confirm the result. Guys, we know that the roots are not always so easy to see by factoring, so this formula will definitely come in handy!

Quadratic Formula Method

The quadratic formula is: x = [-b ± √(b² - 4ac)] / 2a

For our function, a = 3, b = 13, and c = -10. Plugging these values into the formula, we get:

x = [-13 ± √(13² - 4 * 3 * -10)] / (2 * 3)

x = [-13 ± √(169 + 120)] / 6

x = [-13 ± √289] / 6

x = [-13 ± 17] / 6

This gives us two solutions:

x = (-13 + 17) / 6 = 4 / 6 = 2/3

x = (-13 - 17) / 6 = -30 / 6 = -5

As you can see, we got the same roots, x = 2/3 and x = -5, using both methods. This is a great way to double-check your work!

2. Finding the Vertex for f(x) = 3x² + 13x - 10

The vertex is the turning point of the parabola. For a parabola that opens upwards (like this one, since a = 3 is positive), the vertex is the minimum point. The x-coordinate of the vertex can be found using the formula:

x_vertex = -b / 2a

In our case, a = 3 and b = 13, so:

x_vertex = -13 / (2 * 3) = -13/6

To find the y-coordinate of the vertex, we plug this x-value back into the function:

y_vertex = f(-13/6) = 3*(-13/6)² + 13*(-13/6) - 10

y_vertex = 3*(169/36) - 169/6 - 10

y_vertex = 169/12 - 338/12 - 120/12

y_vertex = -289/12

So, the vertex is at (-13/6, -289/12). This is a crucial point for graphing the parabola.

3. Finding the Axis of Symmetry for f(x) = 3x² + 13x - 10

The axis of symmetry 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 simply:

x = x_vertex

Since we found the x-coordinate of the vertex to be -13/6, the axis of symmetry is:

x = -13/6

This line is an important guide when graphing the parabola, as it helps ensure the symmetry of the curve.

4. Finding the Y-intercept for f(x) = 3x² + 13x - 10

The y-intercept is the point where the parabola intersects the y-axis. This occurs when x = 0. So, to find the y-intercept, we simply plug x = 0 into the function:

f(0) = 3*(0)² + 13*(0) - 10

f(0) = -10

So, the y-intercept is at the point (0, -10). This is another key point to plot when graphing the parabola.

5. Graphing f(x) = 3x² + 13x - 10

Now that we've found all the key features, we can graph the function. Here's a summary of what we know:

• Roots (x-intercepts): x = 2/3 and x = -5
• Vertex: (-13/6, -289/12) (approximately (-2.17, -24.08))
• Axis of Symmetry: x = -13/6 (approximately x = -2.17)
• Y-intercept: (0, -10)

To graph the parabola, follow these steps:

1. Plot the vertex. This is the most important point, as it's the turning point of the parabola.
2. Draw the axis of symmetry. This is a vertical line through the vertex.
3. Plot the roots (x-intercepts). These are where the parabola crosses the x-axis.
4. Plot the y-intercept. This is where the parabola crosses the y-axis.
5. Since parabolas are symmetrical, you can plot additional points by reflecting points across the axis of symmetry. For example, if you plot a point to the right of the axis of symmetry, you can plot a corresponding point the same distance to the left.
6. Connect the points with a smooth U-shaped curve. Remember that the parabola extends infinitely in both directions.

By plotting these points and using the symmetry of the parabola, you can create an accurate graph of f(x) = 3x² + 13x - 10. You'll see a parabola that opens upwards, with its vertex below the x-axis and intersecting the x-axis at x = 2/3 and x = -5. The y-intercept will be at (0, -10). So, there you have it, folks! We've successfully found all the key elements and graphed the quadratic function.

Conclusion

Finding the roots, vertex, axis of symmetry, and y-intercept of a quadratic function allows us to fully understand and visualize its behavior. By mastering these concepts, you can confidently analyze and graph any quadratic function. Remember to practice these steps with different examples to solidify your understanding. Keep exploring the fascinating world of mathematics, guys! This will boost your critical thinking and problem-solving skills, making you a math whiz in no time.