Introduction to Quadratic Functions
A quadratic function is a second-degree polynomial function. Its graph is a curve called a parabola. Depending on the sign of the coefficient a, the parabola opens upwards (if a > 0) or downwards (if a < 0). The highest or lowest point of this parabola is called the vertex. To understand how the domain and range of a function are determined, you can refer to the Domain and Range Basics.
Characteristics:
- The standard form of a quadratic function is 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 symmetric about a vertical line called the axis of symmetry.
- The maximum or minimum value of the quadratic function is given by the y-coordinate of the vertex.
Understanding the Basic Transformations of graphs can help visualize how quadratic functions behave.
Vertex of a Quadratic Function
The vertex of a quadratic function is the highest or lowest point on its graph. For a function given by f(x) = ax2 + bx + c, the vertex (h, k) can be found using the following formulas:
- h = -b/2a
- k = f(h)
Example:
Given the function f(x) = 2x2 - 4x + 1, find the vertex.
Solution:
Using the formula for h: h = -(-4)/2(2) = 1
Now, plug this into the function to find k: k = 2(1)2 - 4(1) + 1 = -1
Thus, the vertex is (1, -1).
Intercepts of a Quadratic Function
Intercepts are the points where the graph of the function crosses the axes.
x-intercepts (or roots)
The x-intercepts are the values of x for which f(x) = 0. They can be found using the quadratic formula: x = (-b ± sqrt(b2-4ac))/2a. For more on solving these types of equations, see Solving Quadratic Equations.
y-intercept
The y-intercept is the value of the function when x = 0. It is simply f(0).
Example:
For the function f(x) = x2 - 3x + 2, find the x and y intercepts.
Solution:
Using the quadratic formula for the x-intercepts: x = (3 ± sqrt(9-8))/2 This gives two solutions: x = 1 and x = 2.
For the y-intercept, plug in x = 0: f(0) = 2 So, the y-intercept is (0, 2).
IB Maths Tutor Tip: Mastering the vertex formula and quadratic formula equips you to efficiently find a quadratic function's key characteristics, crucial for solving real-world and mathematical problems.
Axis of Symmetry
The axis of symmetry is a vertical line that divides the graph of a quadratic function into two symmetrical halves. It passes through the vertex of the parabola. The equation for the axis of symmetry is: x = h Where h is the x-coordinate of the vertex.
Example:
For the function f(x) = -x2 + 6x - 5, find the axis of symmetry.
Solution:
Using the formula for h: h = -6/2(-1) = 3 Thus, the equation of the axis of symmetry is x = 3.
Real-world Applications of Quadratic Functions
Quadratic functions can model various real-world scenarios:
1. Projectile Motion: The trajectory of a thrown object can be modelled by a quadratic function. For instance, the height of a ball thrown upwards or downwards can be represented by a quadratic equation. An example of this application is detailed in the notes on Free Fall and Projectile Motion.
2. Business and Economics: Quadratic functions can represent profit or loss based on the number of items produced or sold.
3. Engineering: Quadratic equations can describe certain electrical properties or represent the behaviour of certain materials under stress.
IB Tutor Advice: Practise applying the quadratic formula and vertex calculations in diverse problems to enhance your speed and accuracy for exams, especially under time constraints.
Example:
A ball is thrown upwards, and its height h in metres after t seconds is modelled by the equation h(t) = -5t2 + 20t. Find the time when the ball reaches its maximum height.
Solution:
The maximum height corresponds to the vertex of the parabola. Using the formula for h: h = -20/2(-5) = 2 Thus, the ball reaches its maximum height after 2 seconds.
For further exploration of quadratic functions and their properties, such as how they relate to rational functions, you may find the Basics of Rational Functions useful in expanding your understanding of mathematical concepts.
FAQ
The value of 'c' in a quadratic function represents the y-intercept of the graph. It's the point where the parabola intersects the y-axis. Specifically, the y-intercept is given by the point (0, c). This means that changing the value of 'c' will move the graph up or down without affecting its shape. For instance, if c is increased, the entire parabola will shift upwards, and if c is decreased, it will shift downwards.
The vertex form of a quadratic function, given by f(x) = a(x - h)2 + k, is particularly useful because it directly provides the vertex (h, k) of the parabola. This form is especially handy when one needs to quickly identify the maximum or minimum value of a function or when graphing the function. By simply looking at the equation, one can determine the vertex without additional calculations. Additionally, the vertex form can make certain transformations and translations of the graph more intuitive.
No, a quadratic function can have at most two x-intercepts. These intercepts are also referred to as the roots or solutions of the quadratic equation. Depending on the discriminant (b2 - 4ac), a quadratic function can have two distinct x-intercepts, one repeated x-intercept, or no x-intercepts at all. Specifically, if the discriminant is positive, there are two distinct solutions; if it's zero, there's one repeated solution; and if it's negative, there are no real solutions, meaning the graph doesn't intersect the x-axis.
The direction in which a parabola opens is determined by the coefficient of the x2 term in the quadratic function. If the coefficient (often represented as 'a') is positive, the parabola will open upwards. Conversely, if the coefficient is negative, the parabola will open downwards. This is a fundamental characteristic of quadratic functions. For instance, in the function f(x) = 3x2 + 2x - 1, the coefficient of x2 is 3, which is positive. Therefore, the parabola for this function will open upwards.
The discriminant of a quadratic function, given by the expression b2 - 4ac, plays a crucial role in determining the nature of the roots of the function. If the discriminant is positive, the quadratic function has two distinct real roots. If it's zero, the function has one repeated real root. And if the discriminant is negative, the function has no real roots, implying that the graph of the function does not intersect the x-axis. The discriminant thus provides valuable insight into the structure and properties of the quadratic function's graph.
Practice Questions
To find the x-intercepts, we need to solve for x when f(x) = 0. Using the quadratic formula: x = (-b ± sqrt(b2-4ac))/2a Plugging in the values a = 6.24, b = -2.41, and c = -8.30, we can compute the two possible values for x. After calculating, we find the x-intercepts to be approximately x1 = 1.12 and x2 = -1.18.
The axis of symmetry for a quadratic function is given by the formula: x = -b/2a Using the values a = 6.24 and b = -2.41, we can compute the axis of symmetry. After calculating, the axis of symmetry is approximately x = 0.19.
