Quadratic Function

Interesting Facts About Quadratic Functions:

* Applications of Quadratic functions can be used in many real-world situations. These include: Throwing a ball, shooting a cannon, driving from a platform and hitting a golf ball In sum, quadratic functions are primarily used to find the curve that objects take when they fly through the air..

Graphical/Pictorial

Below, is an example of a quadratic function and how the x-intercept, y-intercept, Equation of axis, and the vertex are found. By finding the X and Y intercepts, one can then interpret the Domain, range, and roots.

Tabular

Below, is a tabular example of a quadratic relationship represented by a table of values. In this quadratic function, y=x^2, when we increase the x−value by one, the y value increases by different values. However, it increases at a constant rate, so the difference of the difference is always 2.

Symbolic

Quadratic functions can be represented symbolically by the equation.

y(x) = ax^2 + bx + c

Where a, b, and c are constants, and a ≠ 0. This form is referred to as standard form. The coefficient a in this form is called the leading coefficient because it is associated with the highest power of x (the squared term).

Verbal

Regardless of what the quadratic function is expressing, whether it be a positive or negative parabolic curve, every quadratic formula shares eight core characteristics.

y = ax^2 + bx + c, where a is not equal to 0
The graph this creates is a parabola. (U-shaped figure)
The parabola will open upward or downward.
A parabola that opens upward contains a vertex that is a minimum point; a parabola that opens downward contains a vertex that is a maximum point.
The domain of a quadratic function consists entirely of real numbers.
If the vertex is a minimum, the range is all real numbers greater than or equal to the y-value. If the vertex is a maximum, the range is all real numbers less than or equal to the y-value.
An axis of symmetry (equation of axis) will divide the parabola into mirror images. The line of symmetry is always a vertical line of the form x = n, where n is a real number, and its axis of symmetry is the vertical line x =0.
The x-intercepts are the points at which a parabola intersects the x-axis. These points are also known as zeroes, roots, solutions, and solution sets. Each quadratic function will have two, one, or no x-intercepts.

By identifying and understanding these core concepts related to quadratic functions, you can use quadratic equations to solve a variety of real-life problems with missing variables and a range of possible solutions.

Examples:

A toy rocket is fired into the air from the top of a barn. Its height (h) above the ground in yards after t seconds is given by the function h(t)=−5t^2+10t+20.
- What was the initial height of the rocket?
- When did the rocket reach its maximum height?

Sketch a graph of the function. Your graphing calculator can be used to produce the graph.

The initial height of the rocket is the height from which it was fired. The time is zero.

h(t)=−5t^2+10t+20
h(t)=−5(0)^2+10(0)+20
h(t)=20 yd

The initial height of the toy rocket is 20 yards. This is the y-intercept of the graph. The y-intercept of a quadratic function written in general form is the value of 'c'.

The time at which the rocket reaches its maximum height is the x-coordinate of the vertex.

t=−b/2a
t=−10/2(−5)
t=1 sec

It takes the toy rocket 1 second to reach its maximum height.

The product of two consecutive positive odd integers is 195. Find the integers.

Let n represent the first positive odd integer. Let n+2 represent the second positive odd integer. Write an equation to represent the problem.

n(n+2)=195

n^2+2n=195

You can solve this equation with a few different methods. Here, use the quadratic formula.

There was a restriction on the solution presented in the problem. The solution must be an odd positive integer. Therefore, 13 is the solution you can use. The two positive odd integers are 13 and 15.