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

by Admin 56 views
Find b and c from Quadratic Graph: f(x) = 2x^2 + bx + c

Alright, let's dive into how to find the values of b and c in a quadratic function given its graph. This is a common problem in algebra, and understanding the approach can really solidify your knowledge of quadratic functions. So, if you've got a graph of a function in the form f(x) = 2x^2 + bx + c, how do you figure out what b and c are? Let's break it down step-by-step.

Understanding the Quadratic Function

First, let's understand what each part of the quadratic function f(x) = 2x^2 + bx + c represents. The general form of a quadratic function is f(x) = ax^2 + bx + c, where:

  • a determines how "wide" or "narrow" the parabola is and whether it opens upwards or downwards. If a > 0, the parabola opens upwards, and if a < 0, it opens downwards. In our case, a = 2.
  • b affects the position of the parabola's axis of symmetry.
  • c is the y-intercept of the parabola, meaning it’s the point where the parabola crosses the y-axis. This gives us a direct way to find the value of c.

Finding the Value of c

The easiest part to find is usually c, because, as mentioned above, c is simply the y-intercept of the graph. The y-intercept is the point where the graph intersects the y-axis. So, look at your graph and see where the parabola crosses the y-axis. The y-coordinate of that point is the value of c. For example, if the graph crosses the y-axis at the point (0, 3), then c = 3. This is because when x = 0, the function becomes f(0) = 2(0)^2 + b(0) + c = c. Therefore, f(0) gives us the y-coordinate of the y-intercept, which is c.

Finding the Value of b

Finding b is a bit more involved, but still manageable. Here are a couple of common methods:

  1. Using the Vertex Form:

    The vertex form of a quadratic equation is f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. The vertex is the point where the parabola changes direction (either the minimum or maximum point). If you can identify the vertex (h, k) from the graph, you can use it to find b. Expand the vertex form to get it into the standard form and then compare coefficients:

    f(x) = a(x - h)^2 + k = a(x^2 - 2hx + h^2) + k = ax^2 - 2ahx + ah^2 + k

    Comparing this with f(x) = 2x^2 + bx + c, we can see that:

    b = -2ah

    Since we know a = 2, we have b = -4h. So, if you know the x-coordinate h of the vertex, you can easily find b.

  2. Using Another Point on the Graph:

    If you know the coordinates of another point on the graph (x, y) (other than the y-intercept or the vertex), you can plug these values into the original equation f(x) = 2x^2 + bx + c and solve for b. You'll already know the value of c from the y-intercept, so you'll have one equation with one unknown (b).

    For example, let's say you know that the point (1, 5) is on the graph, and you've already determined that c = 3. Then, plug in these values:

    5 = 2(1)^2 + b(1) + 3

    5 = 2 + b + 3

    5 = 5 + b

    b = 0

    So in this case, b = 0.

  3. Using the Axis of Symmetry Formula:

    The axis of symmetry for a quadratic function in the form f(x) = ax^2 + bx + c is given by the formula x = -b / (2a). If you can determine the axis of symmetry from the graph (which is the vertical line that passes through the vertex), you can use this formula to solve for b.

    Since we know a = 2, the formula becomes x = -b / (2 * 2) = -b / 4. If you know the x-coordinate of the axis of symmetry, you can set it equal to -b / 4 and solve for b.

    For example, if the axis of symmetry is x = 1, then:

    1 = -b / 4

    b = -4

Example Problem

Let's walk through an example to make sure we've got this down. Suppose we have a quadratic function f(x) = 2x^2 + bx + c, and its graph shows the following:

  • The parabola intersects the y-axis at (0, -2).
  • The vertex of the parabola is at (1, -4).

Step 1: Find c

The y-intercept is (0, -2), so c = -2.

Step 2: Find b

We know the vertex is (1, -4), so h = 1. Using the formula b = -4h (derived from the vertex form), we have:

b = -4 * 1 = -4

So, b = -4.

Therefore, the quadratic function is f(x) = 2x^2 - 4x - 2.

Tips and Tricks

  • Always start with c: Finding the y-intercept is usually the quickest way to get started.
  • Look for the vertex: The vertex provides a lot of information and can help you find b easily.
  • Use additional points: If you have the coordinates of another point on the graph, use it to create an equation and solve for b.
  • Double-check your work: After finding b and c, plug them back into the equation and see if it matches the graph. This can help you catch any mistakes.

Common Mistakes to Avoid

  • Incorrectly identifying the y-intercept: Make sure you're looking at the point where the graph actually crosses the y-axis.
  • Confusing the vertex coordinates: Remember that h is the x-coordinate of the vertex, not the y-coordinate.
  • Algebra mistakes: Be careful when solving equations. A small mistake can lead to the wrong value for b.

Practice Problems

  1. A quadratic function f(x) = 2x^2 + bx + c has a y-intercept at (0, 5) and a vertex at (-1, 3). Find the values of b and c.
  2. The graph of f(x) = 2x^2 + bx + c passes through the point (2, 10) and has a y-intercept at (0, 2). Find b and c.

By working through these problems, you'll become more comfortable with finding the values of b and c from a quadratic function's graph.

Conclusion

Finding the values of b and c in a quadratic function f(x) = 2x^2 + bx + c from its graph 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, along with other points on the graph, can be used to find b. By understanding these relationships and practicing with example problems, you can master this skill and confidently tackle similar problems in algebra.

So, the next time you're faced with a graph and asked to find b and c, remember these steps, and you'll be well on your way to solving the problem! Happy graphing, guys!