Determine The Quadratic Function Whose Graph Is Given Below

Determining the Quadratic Function from its Graph

Determining the quadratic function from its graph involves using the key features of a parabola to construct the equation in the form f(x) = ax² + bx + c or its vertex form f(x) = a(x-h)² + k. This process leverages the understanding of parabolas, their symmetry, vertex, intercepts, and the impact of the 'a' value on the parabola's orientation and width. This article will guide you through various methods, emphasizing the importance of accurately identifying crucial graphical features.

Understanding the Fundamentals of Quadratic Functions

Before delving into the methods, let's refresh our understanding of quadratic functions and their graphical representation.

A quadratic function is a polynomial function of degree two, generally expressed as:

f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants and 'a' ≠ 0.

The graph of a quadratic function is a parabola, a U-shaped curve. The key features of a parabola include:

• Vertex: The highest or lowest point on the parabola. The x-coordinate of the vertex is given by -b/(2a).
• Axis of Symmetry: A vertical line passing through the vertex, dividing the parabola into two symmetrical halves. Its equation is x = -b/(2a).
• x-intercepts (roots or zeros): The points where the parabola intersects the x-axis (where f(x) = 0). These can be found by factoring the quadratic equation or using the quadratic formula: x = [-b ± √(b² - 4ac)] / 2a.
• y-intercept: The point where the parabola intersects the y-axis (where x = 0). This is simply the value of 'c' in the equation f(x) = ax² + bx + c.
• Concavity: The direction the parabola opens. If 'a' > 0, the parabola opens upwards (concave up), and if 'a' < 0, it opens downwards (concave down).

Methods for Determining the Quadratic Function

Several approaches can be used to determine the quadratic function from its graph. The best method depends on the information readily available from the graph.

Method 1: Using the Vertex and Another Point

If the vertex (h, k) and another point (x₁, y₁) on the parabola are known, we can use the vertex form of the quadratic equation:

f(x) = a(x - h)² + k

1. Substitute the vertex coordinates: Replace 'h' and 'k' with the x and y coordinates of the vertex.
2. Substitute the coordinates of the other point: Replace 'x' and 'f(x)' with the coordinates of the known point (x₁, y₁).
3. Solve for 'a': Solve the resulting equation for the constant 'a'.
4. Write the equation: Substitute the values of 'a', 'h', and 'k' back into the vertex form to obtain the quadratic function.

Example:

Let's say the vertex is (2, -1) and another point on the parabola is (0, 3).

1. Vertex form: f(x) = a(x - 2)² - 1
2. Substitute (0, 3): 3 = a(0 - 2)² - 1
3. Solve for 'a': 4 = 4a => a = 1
4. Equation: f(x) = (x - 2)² - 1 or, expanding, f(x) = x² - 4x + 3

Method 2: Using the x-intercepts and Another Point

If the x-intercepts (x₁, 0) and (x₂, 0) and another point (x₃, y₃) are known, we can use the intercept form of the quadratic equation:

f(x) = a(x - x₁)(x - x₂)

1. Substitute the x-intercepts: Replace x₁ and x₂ with the x-coordinates of the intercepts.
2. Substitute the coordinates of the other point: Replace 'x' and 'f(x)' with the coordinates of the known point (x₃, y₃).
3. Solve for 'a': Solve the resulting equation for the constant 'a'.
4. Write the equation: Substitute the values of 'a', x₁, and x₂ back into the intercept form to obtain the quadratic function.

Example:

Suppose the x-intercepts are (-1, 0) and (3, 0), and another point is (1, -4).

1. Intercept form: f(x) = a(x + 1)(x - 3)
2. Substitute (1, -4): -4 = a(1 + 1)(1 - 3)
3. Solve for 'a': -4 = -4a => a = 1
4. Equation: f(x) = (x + 1)(x - 3) or, expanding, f(x) = x² - 2x - 3

Method 3: Using Three Points

If three points (x₁, y₁), (x₂, y₂), and (x₃, y₃) on the parabola are known, we can use the standard form of the quadratic equation:

f(x) = ax² + bx + c

1. Substitute each point: Substitute the coordinates of each point into the equation f(x) = ax² + bx + c, creating a system of three linear equations with three unknowns (a, b, c).
2. Solve the system of equations: Solve the system of equations using techniques like substitution, elimination, or matrices to find the values of a, b, and c.
3. Write the equation: Substitute the values of a, b, and c into the standard form to obtain the quadratic function.

Example:

Let's assume the points are (1, 2), (2, 3), and (3, 6).

This yields the system:

• a + b + c = 2
• 4a + 2b + c = 3
• 9a + 3b + c = 6

Solving this system (using methods beyond the scope of a concise explanation here, but readily solvable using substitution, elimination, or matrix methods) would give the values of a, b, and c, leading to the final quadratic equation.

Interpreting the 'a' Value

The value of 'a' in the quadratic function plays a crucial role in determining the parabola's characteristics:

• Magnitude of 'a': A larger absolute value of 'a' indicates a narrower parabola (steeper curve), while a smaller absolute value indicates a wider parabola (flatter curve).
• Sign of 'a': A positive 'a' indicates a parabola that opens upwards (concave up), while a negative 'a' indicates a parabola that opens downwards (concave down).

Handling Special Cases

Certain situations require adjustments to the methods described above:

• Parabola touching the x-axis: If the parabola touches the x-axis (only one x-intercept), it means the discriminant (b² - 4ac) is zero. In this case, the vertex lies on the x-axis. You can use the vertex and another point method or the three-point method.

• Parabola entirely above or below the x-axis: If the parabola doesn't intersect the x-axis, it means the discriminant is negative. The x-intercepts are complex numbers, and you’ll need to use either the vertex and another point method or the three-point method. The y-intercept can also be helpful in these situations.

Importance of Accuracy in Graph Reading

The accuracy of the determined quadratic function heavily relies on the precision with which you read the coordinates from the graph. Any slight errors in reading the coordinates can lead to significant inaccuracies in the resulting equation. Use a ruler or other tools to assist with accurate measurement.

Conclusion

Determining the quadratic function from its graph is a fundamental skill in algebra and calculus. By understanding the properties of parabolas and applying the appropriate methods, one can accurately derive the equation. Remember that the choice of method depends on the information available from the graph. Accuracy in reading the coordinates from the graph is paramount for achieving a correct result. Practice with diverse examples will reinforce your understanding and skill in this process. The methods described, alongside a clear understanding of quadratic function properties, offer a robust toolkit for tackling various scenarios.