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Finding a quadratic function given vertex and another point
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Finding a Quadratic Function Given Vertex and Another Point
Introduction
Understanding how to determine a quadratic function based on its vertex and an additional point is fundamental in mastering quadratic equations within the Cambridge IGCSE Mathematics curriculum. This skill not only reinforces the conceptual grasp of parabolas but also enhances problem-solving abilities essential for higher-level mathematics and real-world applications.
Key Concepts
1. Understanding Quadratic Functions
A quadratic function is a second-degree polynomial with the general form:
$$
f(x) = ax^2 + bx + c
$$
where \(a\), \(b\), and \(c\) are constants, and \(a \neq 0\). The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of \(a\).
The vertex of a parabola is its highest or lowest point, depending on whether it opens downward or upward, respectively. The vertex form of a quadratic function is given by:
$$
f(x) = a(x - h)^2 + k
$$
where \((h, k)\) is the vertex of the parabola. This form is particularly useful when the vertex is known, as it simplifies the process of writing the equation of the quadratic function.
2. Converting Between Forms
To find a quadratic function when given the vertex and another point, it's often convenient to use the vertex form. Converting from standard form to vertex form involves completing the square, which reorganizes the equation to highlight the vertex.
For example, starting with the standard form:
$$
f(x) = ax^2 + bx + c
$$
Completing the square involves rewriting the quadratic and linear terms as a perfect square trinomial:
$$
f(x) = a\left(x^2 + \frac{b}{a}x\right) + c
$$
$$
f(x) = a\left(\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2}\right) + c
$$
$$
f(x) = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c
$$
Comparing this with the vertex form allows us to identify the vertex \((h, k)\) as:
$$
h = -\frac{b}{2a}, \quad k = c - \frac{b^2}{4a}
$$
3. Determining the Quadratic Function Using Vertex and a Point
When provided with the vertex \((h, k)\) and another point \((x_1, y_1)\) on the parabola, the vertex form is instrumental in constructing the quadratic function.
Starting with the vertex form:
$$
f(x) = a(x - h)^2 + k
$$
Substituting the coordinates of the additional point into the equation allows the determination of the coefficient \(a\):
$$
y_1 = a(x_1 - h)^2 + k
$$
Solving for \(a\):
$$
a = \frac{y_1 - k}{(x_1 - h)^2}
$$
Once \(a\) is known, the complete quadratic function can be written in vertex form.
4. Example Problem
**Problem:** Find the quadratic function with vertex at \( (2, 3) \) and passing through the point \( (4, 7) \).
**Solution:**
Given:
- Vertex \((h, k) = (2, 3)\)
- Point \((x_1, y_1) = (4, 7)\)
Start with the vertex form:
$$
f(x) = a(x - h)^2 + k
$$
Substitute the known vertex:
$$
f(x) = a(x - 2)^2 + 3
$$
Now, substitute the point \((4, 7)\) to find \(a\):
$$
7 = a(4 - 2)^2 + 3
$$
$$
7 = a(2)^2 + 3
$$
$$
7 = 4a + 3
$$
Subtract 3 from both sides:
$$
4a = 4
$$
Divide by 4:
$$
a = 1
$$
Thus, the quadratic function is:
$$
f(x) = 1(x - 2)^2 + 3
$$
Simplified:
$$
f(x) = (x - 2)^2 + 3
$$
Expanding to standard form:
$$
f(x) = x^2 - 4x + 7
$$
5. Graphical Interpretation
Plotting the points and the vertex on a coordinate plane provides a visual understanding of the quadratic function. The vertex \((2, 3)\) signifies the parabola's turning point, and the additional point \((4, 7)\) ensures that the parabola passes through that specific location, confirming the accuracy of the function derived.
6. Practical Applications
Quadratic functions are utilized in various real-world scenarios, including projectile motion, economics for profit maximization, and engineering for designing parabolic structures. Understanding how to derive these functions from specific points and vertices is crucial for modeling and solving practical problems.
Advanced Concepts
1. Deriving the Vertex Formula
The vertex form of a quadratic function is not just a manipulation of the standard form but is rooted in the properties of parabolas. Deriving the vertex formula involves completing the square, which transforms the standard quadratic equation into a form that clearly reveals the vertex coordinates.
Starting with the standard form:
$$
f(x) = ax^2 + bx + c
$$
Complete the square to convert it into vertex form:
$$
f(x) = a\left(x^2 + \frac{b}{a}x\right) + c
$$
$$
f(x) = a\left(\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2}\right) + c
$$
$$
f(x) = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c
$$
Thus, the vertex \((h, k)\) is:
$$
h = -\frac{b}{2a}, \quad k = c - \frac{b^2}{4a}
$$
This derivation provides a deeper understanding of how the coefficients \(a\), \(b\), and \(c\) influence the position of the vertex on the graph.
2. Analyzing the Impact of Coefficient \(a\)
The coefficient \(a\) in the quadratic function plays a pivotal role in determining the parabola's width and direction. Specifically:
- If \(a > 0\), the parabola opens upwards.
- If \(a < 0\), the parabola opens downwards.
- The absolute value of \(a\) affects the "stretch" or "compression" of the parabola. A larger \(|a|\) results in a narrower parabola, while a smaller \(|a|\) produces a wider one.
Understanding the influence of \(a\) is essential when interpreting the quadratic function's graph and in applications where the shape of the parabola is critical.
3. Discriminant and Its Relation to Roots
The discriminant of a quadratic equation, given by \(D = b^2 - 4ac\), provides valuable information about the nature of the roots of the equation:
- If \(D > 0\): Two distinct real roots exist.
- If \(D = 0\): One real root exists (the vertex touches the x-axis).
- If \(D < 0\): No real roots exist; the roots are complex.
When the quadratic function is in vertex form, the discriminant can still be determined by expanding the equation to standard form, allowing for analysis of the parabola's intersection with the x-axis.
4. Symmetry of the Parabola
Parabolas exhibit symmetry about a vertical line passing through their vertex, known as the axis of symmetry. For the vertex \((h, k)\), the axis of symmetry is the line \(x = h\). This property is crucial in solving quadratic equations and optimizing functions, as it allows for predicting the parabola's behavior on either side of the vertex.
5. Optimization Problems Involving Quadratic Functions
Quadratic functions are frequently employed in optimization problems where a maximum or minimum value needs to be determined. For instance, determining the maximum height of a projectile or the minimum cost in a business scenario can be modeled using quadratic functions. Utilizing the vertex form simplifies finding these optimal values, as the vertex directly provides the maximum or minimum point of the function.
6. Interdisciplinary Connections
Quadratic functions bridge various disciplines, demonstrating their versatility and significance. In physics, they model projectile motion under uniform gravity. In economics, they represent cost and revenue functions to find profit maximization. Engineering utilizes quadratic functions in designing arches and structures that require specific load distributions. Understanding quadratic functions enhances problem-solving skills across these diverse fields.
Comparison Table
| Aspect | Standard Form | Vertex Form |
| General Equation | $f(x) = ax^2 + bx + c$ | $f(x) = a(x - h)^2 + k$ |
| Vertex Identification | Requires completing the square or using $h = -\frac{b}{2a}$, $k = f(h)$ | Vertex is directly given as $(h, k)$ |
| Graph Characteristics | Less intuitive for graphing vertex | Facilitates easy graphing by highlighting vertex |
| Usefulness | Standard for general quadratic equations | Ideal when vertex information is known |
| Conversion | Can be converted to vertex form by completing the square | Can be converted to standard form by expanding the equation |
Summary and Key Takeaways
- Quadratic functions can be effectively determined using the vertex form when given the vertex and an additional point.
- The vertex form simplifies the process of identifying the vertex and graphing the parabola.
- Understanding the role of coefficient \(a\) is crucial for determining the direction and width of the parabola.
- Advanced concepts like the discriminant and symmetry enhance the analytical capabilities in solving quadratic equations.
- Quadratic functions have wide-ranging applications across various disciplines, highlighting their importance in real-world problem-solving.
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Examiner Tip
Tips
Always Double-Check Your Vertex: Ensure you correctly calculate the vertex coordinates before proceeding.
Use Mnemonics: Remember "A squared" for \(a(x - h)^2 + k\) to link the form to its vertex.
Practice Graphing: Regularly sketch parabolas to visualize the relationship between the vertex and other points.
Review the Discriminant: Understanding the discriminant helps predict the number of real roots, aiding in problem-solving.
Did You Know
Did You Know
Did you know that the concept of quadratic functions dates back to ancient Babylonian mathematics, where they were used to solve area problems? Additionally, quadratic equations play a crucial role in modern technology, such as in the algorithms behind computer graphics and animation. Understanding quadratic functions not only helps in academic success but also in various technological advancements that shape our daily lives.
Common Mistakes
Common Mistakes
Mistake 1: Incorrectly identifying the vertex coordinates when converting from standard to vertex form.
Incorrect: Assuming the vertex is at \((b/2a, c)\).
Correct: Use \(h = -\frac{b}{2a}\) and \(k = f(h)\) to find the vertex \((h, k)\).
Mistake 2: Forgetting to square the difference when substituting the additional point into the vertex form.
Incorrect: \(a = \frac{y_1 - k}{x_1 - h}\).
Correct: \(a = \frac{y_1 - k}{(x_1 - h)^2}\).
FAQ
What is the vertex form of a quadratic function?
The vertex form is \(f(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola.
How do you find the coefficient \(a\) in the vertex form?
Substitute the coordinates of the additional point into the vertex form equation and solve for \(a\).
Can a quadratic function have more than one vertex?
No, a quadratic function has exactly one vertex, which is the highest or lowest point on its graph.
What happens to the graph if the coefficient \(a\) is negative?
If \(a\) is negative, the parabola opens downward, making the vertex the highest point on the graph.
How do you convert the vertex form to the standard form?
Expand the equation \(f(x) = a(x - h)^2 + k\) to get \(f(x) = ax^2 + bx + c\).
1.
Number
2.
Statistics
3.
Algebra
4.
Transformations and Vectors
5.
Geometry
6.
Functions
7.
Mensuration
8.
Coordinate Geometry
9.
Trigonometry
10.
Probability
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