Find B & C From Graph F(x) = 2x^2 + Bx + C

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Find b & c from Graph f(x) = 2x^2 + bx + c

Alright, guys, let's dive into how to extract the values of b and c from the graph of a quadratic function in the form f(x) = 2x² + bx + c. This is a classic problem that combines graphical analysis with algebraic understanding. We're going to break it down step-by-step, making it super easy to follow. So, grab your thinking caps, and let's get started!

Understanding the Quadratic Function

First, let's get a handle on what the different parts of our quadratic function f(x) = 2x² + bx + c actually mean. The general form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are constants. In our case, a = 2. The coefficient a determines the direction and width of the parabola. If a is positive (like our 2), the parabola opens upwards. If a is negative, it opens downwards. The larger the absolute value of a, the narrower the parabola.

The coefficient b affects the position of the parabola's axis of symmetry. The axis of symmetry is a vertical line that passes through the vertex of the parabola, splitting it into two symmetrical halves. The x-coordinate of the vertex is given by the formula x = -b / 2a. This is a crucial piece of information that we'll use later.

Lastly, c is the y-intercept of the parabola. This is the point where the parabola intersects the y-axis. In other words, it's the value of f(x) when x = 0. So, f(0) = a(0)² + b(0) + c = c. This means that the y-intercept is simply the value of c. This is super handy because we can read the y-intercept directly from the graph!

Knowing these basics, we can approach the problem strategically. The key is to identify key features of the graph, such as the vertex and the y-intercept, and use these features to solve for b and c.

Identifying Key Features from the Graph

Now, let's talk about how to extract the necessary information from the graph. When you're given a graph of a quadratic function, there are a few key features you'll want to identify:

  1. The Vertex: The vertex 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 coordinates of the vertex are often written as (h, k), where h is the x-coordinate and k is the y-coordinate. The x-coordinate of the vertex is also the axis of symmetry.
  2. The Y-Intercept: As we mentioned earlier, the y-intercept is the point where the parabola intersects the y-axis. This is the point where x = 0. The y-coordinate of this point is the value of c.
  3. Any Other Points: If possible, try to identify any other points on the graph that have integer coordinates. These points can be used to create equations that help you solve for b and c.

Once you've identified these features, you can use them to set up a system of equations and solve for b and c. Let's see how this works with an example.

Using the Y-Intercept to Find C

The easiest value to find is usually c because, as we discussed, c is simply the y-intercept of the graph. Look at the graph and see where the parabola crosses the y-axis. The y-coordinate of that point is your value for c. For example, if the graph crosses the y-axis at the point (0, 3), then c = 3. It's that simple!

So, in our function f(x) = 2x² + bx + c, if you see that the graph intersects the y-axis at, say, y = -4, you immediately know that c = -4. This gives you a great starting point for figuring out the rest of the function.

Once you have the value of c, you can plug it back into the equation, which simplifies the problem a bit. Now you only have one unknown variable, b, to solve for.

Using the Vertex to Find B

Finding b is a bit more involved but still manageable. Remember that the x-coordinate of the vertex is given by the formula x = -b / 2a. Since we know the value of a (which is 2 in our case), and we can determine the x-coordinate of the vertex from the graph, we can solve for b.

Here's how:

  1. Identify the Vertex: Look at the graph and find the coordinates of the vertex. Let's say the vertex is at the point (1, -2).
  2. Use the Vertex Formula: We know that the x-coordinate of the vertex is x = -b / 2a. In our example, the x-coordinate is 1, and a = 2. So we have the equation 1 = -b / (2 * 2).
  3. Solve for B: Simplify the equation: 1 = -b / 4. Multiply both sides by 4: 4 = -b. Finally, multiply by -1: b = -4.

So, in this example, we found that b = -4. Now we have both b and c, and we've completely determined the quadratic function from its graph.

Example: Putting It All Together

Let's walk through a complete example to solidify our understanding. Suppose we have a graph of the function f(x) = 2x² + bx + c, and we observe the following:

  • The y-intercept is at (0, 5).
  • The vertex is at (2, -3).

Here's how we would find b and c:

  1. Find C: The y-intercept is (0, 5), so c = 5.
  2. Find B: The vertex is (2, -3), so the x-coordinate of the vertex is 2. Using the formula x = -b / 2a, we have 2 = -b / (2 * 2). Simplify: 2 = -b / 4. Multiply both sides by 4: 8 = -b. Multiply by -1: b = -8.

Therefore, b = -8 and c = 5. Our quadratic function is f(x) = 2x² - 8x + 5.

Additional Tips and Tricks

  • Double-Check Your Work: After finding b and c, plug them back into the equation and see if the resulting function matches the graph. You can do this by plugging in a few x-values and comparing the calculated y-values with the graph.
  • Use Additional Points: If you have another point on the graph (other than the vertex and y-intercept), you can use it to create another equation. This can be helpful if you're unsure about the accuracy of your vertex or y-intercept readings.
  • Be Careful with Signs: Pay close attention to the signs of the coordinates, especially when using the vertex formula. A simple sign error can throw off your entire calculation.

Conclusion

Finding the values of b and c from the graph of a quadratic function f(x) = 2x² + bx + c involves identifying key features such as the y-intercept and the vertex. The y-intercept gives you the value of c directly, while the vertex, combined with the formula x = -b / 2a, allows you to solve for b. With a bit of practice and attention to detail, you'll be able to tackle these problems with confidence. Keep practicing, and you'll master this skill in no time! And remember, understanding the underlying concepts is just as important as memorizing the formulas. Happy graphing, guys!