Find B And C From Quadratic Graph: F(x) = 2x^2 + Bx + C

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Find b and c from Quadratic Graph: f(x) = 2x^2 + bx + c

Alright, let's dive into figuring out 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 common type of problem in algebra, and with a few key concepts, it becomes quite manageable. So, buckle up, and let's get started!

Understanding the Quadratic Function

Before we jump into the specifics, let's make sure we're all on the same page with the basics of a quadratic function. A quadratic function is a polynomial function of degree two, generally written as 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, which is a U-shaped curve. The coefficient a determines whether the parabola opens upwards (a > 0) or downwards (a < 0). In our case, a = 2, so the parabola opens upwards.

The coefficient 'b' affects the position of the parabola's axis of symmetry, and 'c' represents the y-intercept of the parabola. The vertex of the parabola is the point where the parabola changes direction; it's either the minimum point (if a > 0) or the maximum point (if a < 0). Understanding these properties is crucial for solving our problem.

Identifying Key Features from the Graph

When you're given the graph of a quadratic function, there are a few key features that you should immediately look for. These features will provide the necessary information to determine the values of 'b' and 'c'.

  1. Y-intercept: The y-intercept is the point where the parabola intersects the y-axis. This point has the coordinates (0, y), and the y-coordinate is the value of 'c'. So, if you can identify the y-intercept from the graph, you immediately know the value of 'c'.

  2. Vertex: The vertex is the highest or lowest point on the parabola. The coordinates of the vertex are usually denoted as (h, k). The x-coordinate of the vertex, h, is related to the coefficients a and b by the formula h = -b / (2a). Since we know the value of a (in our case, a = 2), if we can find the x-coordinate of the vertex from the graph, we can use this formula to solve for 'b'.

  3. Additional Points: If the y-intercept and vertex aren't enough to determine 'b' and 'c', look for any other points on the graph whose coordinates you can easily read. Each point (x, y) on the graph satisfies the equation f(x) = 2x² + bx + c. If you know the coordinates of such a point, you can plug them into the equation to get a linear equation in terms of 'b' and 'c'. You can then solve this equation simultaneously with any other equations you have involving 'b' and 'c'.

Step-by-Step Solution

Let's break down the process into a step-by-step solution. Suppose we have the graph of the function f(x) = 2x² + bx + c.

Step 1: Find the y-intercept

Locate the point where the parabola intersects the y-axis. Let's say the y-intercept is at the point (0, 4). This means that when x = 0, f(x) = 4. Plugging these values into the equation, we get:

f(0) = 2(0)² + b(0) + c = 4

This simplifies to:

c = 4

So, we've found the value of 'c'!

Step 2: Find the Vertex

Next, we need to find the coordinates of the vertex. Let's say the vertex is at the point (1, 2). This means that when x = 1, f(x) = 2. Also, the x-coordinate of the vertex, h, is 1. We can use the formula h = -b / (2a) to find 'b'. Since a = 2 and h = 1, we have:

1 = -b / (2 * 2)

1 = -b / 4

Multiplying both sides by 4, we get:

4 = -b

So,

b = -4

Step 3: Verify the Solution

To make sure we've got the correct values for 'b' and 'c', we can plug them back into the equation and see if the vertex and y-intercept match the graph. Our equation is now:

f(x) = 2x² - 4x + 4

Let's check the vertex. We know that the x-coordinate of the vertex is x = -b / (2a) = -(-4) / (2 * 2) = 4 / 4 = 1. Now, let's find the y-coordinate of the vertex by plugging x = 1 into the equation:

f(1) = 2(1)² - 4(1) + 4 = 2 - 4 + 4 = 2

So, the vertex is indeed at (1, 2), which matches the graph. We already know that the y-intercept is at (0, 4), which also matches the graph. Therefore, our values for 'b' and 'c' are correct.

Alternative Method: Using Another Point

If, for some reason, you can't easily identify the vertex, you can use another point on the graph. Let's say we know that the point (2, 4) lies on the graph. We already know that c = 4 from the y-intercept. Now, we can plug the coordinates of the point (2, 4) into the equation:

f(2) = 2(2)² + b(2) + 4 = 4

This simplifies to:

8 + 2b + 4 = 4

2b + 12 = 4

2b = -8

b = -4

As you can see, we get the same value for 'b' using this method.

Common Mistakes to Avoid

  • Incorrectly Identifying the Vertex: Make sure you accurately identify the coordinates of the vertex from the graph. A small error here can lead to incorrect values for 'b'.
  • Algebra Mistakes: Be careful with your algebra when solving for 'b' and 'c'. Double-check your calculations to avoid simple errors.
  • Misreading the Graph: Ensure that you correctly read the coordinates of the y-intercept and any other points you use from the graph.

Practice Problems

To solidify your understanding, try solving a few practice problems. Here's one to get you started:

Problem: The graph of the function f(x) = 2x² + bx + c has a y-intercept at (0, -2) and a vertex at (1, -4). Find the values of 'b' and 'c'.

Solution:

  1. Find c: The y-intercept is (0, -2), so c = -2.
  2. Find b: The vertex is at (1, -4), so h = 1. Using the formula h = -b / (2a), we have 1 = -b / (2 * 2), which simplifies to b = -4.

So, the values are b = -4 and c = -2.

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. By using the y-intercept to find 'c' and the vertex to find 'b', or by using additional points on the graph, you can solve for these coefficients. Remember to be careful with your algebra and double-check your answers to avoid common mistakes. With a bit of practice, you'll become proficient at solving these types of problems. Keep up the great work, and you'll master these concepts in no time! Guys, you got this! Understanding these fundamental concepts of quadratic functions will not only help you in your algebra class but also provide a solid foundation for more advanced topics in mathematics and beyond. So, keep exploring, keep practicing, and never stop learning! Now go out there and conquer those quadratic graphs!