Quadratic functions are second-degree polynomials that are fundamental in algebra and play a pivotal role in various mathematical and real-world applications. Their graphs, known as parabolas, have distinctive shapes that can open upwards or downwards. This section delves deep into the intricacies of graphing quadratic functions, highlighting their key features: the vertex, intercepts, axis of symmetry, and the discriminant.
Introduction to Quadratic Functions
A quadratic function is typically represented in the form: y = ax2 + bx + c Where:
- a is the coefficient of x2 and determines the direction in which the parabola opens. If a > 0, the parabola opens upwards, and if a < 0, it opens downwards.
- b is the coefficient of x and influences the position and direction of the parabola.
- c is the constant term and determines the y-intercept of the graph.
Vertex
The vertex is a significant point on the parabola. It can be the highest or lowest point, depending on the sign of a.
- The x-coordinate of the vertex is given by: x = -b/2a
- The y-coordinate can be found by substituting the x-coordinate into the quadratic equation.
Example: Consider the quadratic function y = 2x2 - 4x + 1. Using the formula, the x-coordinate of the vertex is: x = -(-4)/2(2) = 1 Substituting x = 1 into the equation, we get y = -1. Hence, the vertex is (1, -1).
Intercepts
Intercepts are the points where the graph intersects the axes.
- X-intercepts (Roots or Zeros): These are the x-values for which y = 0. To find them, set y to zero in the quadratic equation and solve for x. The quadratic formula, which is derived from completing the square, can be used: x = (-b ± sqrt(b2 - 4ac))/2a
- Y-intercept: This is the y-value when x = 0. To determine it, set x to zero in the equation.
Example: For y = x2 - 3x + 2: The x-intercepts are found by setting y = 0. Solving, we get x = 1 and x = 2. The y-intercept is found by setting x = 0, giving y = 2. Thus, the y-intercept is (0, 2).
Axis of Symmetry
The axis of symmetry is a vertical line that bisects the parabola, creating two mirror-image halves.
- The equation for the axis of symmetry is: x = -b/2a
Example: For y = -x2 + 4x - 3, the axis of symmetry is: x = 4/2 = 2
Discriminant
The discriminant provides insights into the nature of the roots of a quadratic equation. It is calculated as: Δ = b2 - 4ac
- If Δ > 0, the quadratic has two distinct real roots.
- If Δ = 0, there's one real root (a repeated root).
- If Δ < 0, there are no real roots; they are complex conjugate roots.
Example: For x2 - 4x + 3 = 0, the discriminant is: Δ = (-4)2 - 4(1)(3) = 4 Since Δ > 0, the equation has two distinct real roots.
Practical Applications
Quadratic functions are ubiquitous in various fields:
- Physics: Quadratics can describe the motion of objects under gravity.
- Economics: They can represent profit and loss functions in business scenarios.
- Engineering: Quadratics can model certain stress-strain relationships in materials.
Practice Question: Given the quadratic function y = x2 - 6x + 8, determine the vertex, axis of symmetry, and x-intercepts.
Solution: Using the formulas mentioned above, you can determine that the vertex is (3, -1), the axis of symmetry is x = 3, and the x-intercepts are x = 2 and x = 4.
FAQ
The discriminant, given by b2 - 4ac, plays a crucial role in determining the nature of the roots of a quadratic equation. In real-world applications, the discriminant can provide insights into the feasibility or number of solutions to a problem. For instance, in physics, when calculating the time an object takes to hit the ground, a negative discriminant might indicate that there's no real-time solution, perhaps due to incorrect data. Similarly, in finance or economics, the discriminant can indicate possible break-even points or the number of solutions to an optimisation problem.
No, a quadratic function can have at most two x-intercepts. The x-intercepts, also known as roots or zeros, are the points where the graph of the function intersects the x-axis. Since a quadratic is a second-degree polynomial, it can have up to two real roots. However, depending on the discriminant, it might have two distinct real roots, one repeated real root, or no real roots at all (in which case it would have two complex conjugate roots). The discriminant (b2 - 4ac) determines the nature of these roots.
The value of 'a' in the quadratic function y = ax2 + bx + c directly affects the width and direction of the parabola. If the absolute value of 'a' is large, the parabola will be narrower, and if the absolute value of 'a' is small (but not zero), the parabola will be wider. Essentially, 'a' determines the "stretch" or "compression" of the parabola. A positive 'a' value means the parabola opens upwards, while a negative 'a' value means it opens downwards. By changing the value of 'a', one can control how spread out or steep the curve of the parabola is.
The nature of the extremum (maximum or minimum) of a quadratic function is determined by the sign of the coefficient 'a'. If 'a' is positive, the parabola opens upwards, and the vertex represents the minimum value of the function. Conversely, if 'a' is negative, the parabola opens downwards, and the vertex represents the maximum value. This is crucial in optimisation problems where one might need to find the highest or lowest value of a function under given constraints. For instance, in business, it can help determine the optimal price to charge to maximise revenue.
The graph of a quadratic function is always a parabola due to the degree of the polynomial. A quadratic function is a second-degree polynomial, and the highest power of the variable is 2. This results in a curve that is symmetric about its axis of symmetry. The coefficient of the x2 term determines the direction of the parabola. If the coefficient is positive, the parabola opens upwards, and if it's negative, it opens downwards. The nature of the squared term ensures that as x moves away from the vertex in both directions, y values increase or decrease quadratically, giving the characteristic U-shape.
Practice Questions
The vertex of the quadratic function can be found using the formula for the x-coordinate of the vertex: x = -b/2a. For the given function, a = 3 and b = -12. Plugging these values in, we get: x = 12/6 = 2 Substituting x = 2 into the original equation, we get the y-coordinate of the vertex: y = 3(22) - 12(2) + 7 = -5 Thus, the vertex is (2, -5).
The axis of symmetry is a vertical line passing through the vertex, so its equation is x = 2.
For the x-intercepts, we set y = 0 and solve for x. Using the quadratic formula or factoring, we can determine the x-intercepts. For this function, the x-intercepts are approximately x = 0.897 and x = 2.603.
To determine the maximum height reached by the ball, we need to find the vertex of the parabola represented by the equation h(t) = -5t2 + 14t + 2. The x-coordinate (or in this context, the time t) of the vertex is given by: t = -b/2a For the given function, a = -5 and b = 14. Plugging these values in, we get: t = 14/10 = 1.4 This means the ball reaches its maximum height after 1.4 seconds.
Substituting t = 1.4 into the equation, we get the maximum height: h(1.4) = -5(1.42) + 14(1.4) + 2 = 11.8 Thus, the maximum height reached by the ball is approximately 11.8 metres.
Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.