Graphing F(x) = X^2 - 4x + 7 Finding Vertex And Plotting Parabola

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In this comprehensive guide, we will delve into the process of graphing the quadratic function f(x) = x^2 - 4x + 7. Quadratic functions, characterized by their parabolic shapes, are fundamental in mathematics and have wide-ranging applications in physics, engineering, and economics. To accurately graph this function, we will first identify and plot the vertex, which is the parabola's turning point. Subsequently, we will determine and plot a second point to establish the parabola's orientation and width. By following these steps, we can effectively visualize the behavior of the quadratic function and gain a deeper understanding of its properties.

1. Understanding Quadratic Functions

Before we dive into graphing, let's briefly review the characteristics of quadratic functions. A quadratic function is a polynomial function of degree two, generally expressed in the form:

f(x) = ax^2 + 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 symmetrical U-shaped curve. The parabola opens upwards if a > 0 and downwards if a < 0. The vertex of the parabola is the point where the function reaches its minimum (if a > 0) or maximum (if a < 0) value. The axis of symmetry is a vertical line passing through the vertex, dividing the parabola into two symmetrical halves.

2. Finding the Vertex of f(x) = x^2 - 4x + 7

The vertex is a crucial point for graphing a parabola, as it provides the parabola's turning point and axis of symmetry. There are two primary methods to determine the vertex: completing the square and using the vertex formula.

2.1. Completing the Square

Completing the square involves rewriting the quadratic function in vertex form:

f(x) = a(x - h)^2 + k

where (h, k) represents the vertex of the parabola. To complete the square for our function f(x) = x^2 - 4x + 7, we follow these steps:

  1. Group the x terms: f(x) = (x^2 - 4x) + 7
  2. Complete the square: Take half of the coefficient of the x term (-4), square it ((-2)^2 = 4), and add and subtract it inside the parentheses: f(x) = (x^2 - 4x + 4 - 4) + 7
  3. Rewrite as a squared term: f(x) = (x - 2)^2 - 4 + 7
  4. Simplify: f(x) = (x - 2)^2 + 3

Now the function is in vertex form, and we can identify the vertex as (2, 3). This vertex represents the minimum point of the parabola, as the coefficient of the squared term is positive (a = 1).

2.2. Using the Vertex Formula

The vertex formula provides a direct method to calculate the vertex coordinates. For a quadratic function in the form f(x) = ax^2 + bx + c, the vertex (h, k) is given by:

h = -b / 2a

k = f(h)

For our function f(x) = x^2 - 4x + 7, we have a = 1, b = -4, and c = 7. Applying the vertex formula:

h = -(-4) / (2 * 1) = 2

k = f(2) = (2)^2 - 4(2) + 7 = 4 - 8 + 7 = 3

Thus, the vertex is (2, 3), which confirms the result obtained by completing the square. The vertex formula is a reliable shortcut for finding the vertex, particularly when dealing with more complex quadratic functions.

3. Finding a Second Point

To accurately graph the parabola, we need at least one more point besides the vertex. We can choose any x-value other than the x-coordinate of the vertex and calculate the corresponding y-value. A simple choice is often x = 0, which gives us the y-intercept.

For f(x) = x^2 - 4x + 7, when x = 0:

f(0) = (0)^2 - 4(0) + 7 = 7

So, the second point is (0, 7). This point lies on the parabola and helps define its shape and direction. Alternatively, we could choose another value for x, such as x = 1:

f(1) = (1)^2 - 4(1) + 7 = 1 - 4 + 7 = 4

This gives us another point, (1, 4), which also helps in accurately graphing the function. The key is to select a point that is easy to calculate and provides a clear indication of the parabola's curve.

4. Graphing the Function

Now that we have the vertex (2, 3) and a second point (0, 7), we can graph the function f(x) = x^2 - 4x + 7.

  1. Plot the vertex: Locate the point (2, 3) on the coordinate plane and mark it. This is the turning point of the parabola.
  2. Plot the second point: Locate the point (0, 7) on the coordinate plane and mark it. This point helps determine the parabola's shape and direction.
  3. Draw the axis of symmetry: Draw a vertical dashed line through the vertex (x = 2). This line divides the parabola into two symmetrical halves.
  4. Find the symmetrical point: Since parabolas are symmetrical, we can find a third point by reflecting the point (0, 7) across the axis of symmetry. The point (0, 7) is 2 units to the left of the axis of symmetry (x = 2), so its symmetrical point will be 2 units to the right, at (4, 7). Plot this point.
  5. Sketch the parabola: Draw a smooth, U-shaped curve that passes through the plotted points. The parabola should be symmetrical about the axis of symmetry and open upwards since the coefficient of the x^2 term is positive. The graph provides a visual representation of the function's behavior, illustrating its minimum point and symmetrical nature.

5. Key Features of the Graph

By graphing the function, we can visually identify several key features:

  • Vertex: The vertex (2, 3) is the minimum point of the parabola, indicating the smallest value the function can attain.
  • Axis of symmetry: The vertical line x = 2 divides the parabola into two symmetrical halves.
  • Y-intercept: The point (0, 7) where the parabola intersects the y-axis. This is found by setting x = 0 in the function.
  • Direction of opening: The parabola opens upwards, which is indicated by the positive coefficient of the x^2 term.
  • Domain: The domain of the function is all real numbers, as the parabola extends infinitely in both horizontal directions.
  • Range: The range of the function is y ≥ 3, as the parabola's lowest point is at y = 3 and it extends upwards infinitely. The key features of the graph provide essential information about the function's behavior and characteristics.

6. Applications of Graphing Quadratic Functions

Graphing quadratic functions is not just a mathematical exercise; it has numerous practical applications. For example:

  • Physics: Projectile motion can be modeled using quadratic functions, and graphing the function can help determine the trajectory, maximum height, and range of the projectile.
  • Engineering: Quadratic functions are used to design parabolic reflectors, such as those used in satellite dishes and headlights. The vertex of the parabola is the focal point where the signal or light is concentrated.
  • Economics: Profit and cost functions can often be modeled using quadratic equations. Graphing these functions can help businesses determine the break-even point, maximum profit, and minimum cost.
  • Optimization Problems: Many optimization problems in various fields involve finding the maximum or minimum value of a quadratic function. Graphing the function can provide a visual representation of the problem and help identify the optimal solution.

7. Conclusion

Graphing the quadratic function f(x) = x^2 - 4x + 7 involves several key steps: finding the vertex, determining a second point, and plotting these points to sketch the parabola. The vertex, which can be found by completing the square or using the vertex formula, is the turning point of the parabola. The axis of symmetry, a vertical line through the vertex, divides the parabola into two symmetrical halves. By plotting the vertex and at least one other point, we can accurately graph the quadratic function and visualize its behavior. This process is essential for understanding the properties of quadratic functions and their applications in various fields. The ability to graph quadratic functions is a fundamental skill in mathematics and provides a powerful tool for solving real-world problems.